# Anisotropic refinable functions and the tile B-splines ††thanks:  The second
author is supported by by the Russian Science Foundation (project no.
23-71-30001) at Lomonosov Moscow State University.

Vladimir Yu. Protasov and Tatyana Zaitseva DISIM of University of L’Aquila,
e-mail<EMAIL_ADDRESS>State University e-mail:
<EMAIL_ADDRESS>
###### Abstract

The regularity of refinable functions has been analysed in an extensive
literature and is well-understood in two cases: 1) univariate 2) multivariate
with an isotropic dilation matrix. The general (non-isotropic) case offered a
great resistance. It was done only recently by developing the matrix method.
In this paper we make the next step and extend the Littlewood-Paley type
method, which is very efficient in the aforementioned special cases, to
general equations with arbitrary dilation matrices. This gives formulas for
the higher order regularity in $W_{2}^{k}({\mathbb{R}}^{n})$ by means of the
Perron eigenvalue of a finite-dimensional linear operator on a special cone.
Applying those results to recently introduced tile B-splines, we prove that
they can have a higher smoothness than the classical ones of the same order.
Moreover, the two-digit tile B-splines have the minimal support of the mask
among all refinable functions of the same order of approximation. This proves,
in particular, the lowest algorithmic complexity of the corresponding
subdivision schemes. Examples and numerical results are provided.

Keywords: multivariate wavelets, frames, refinable function, Hölder
regularity, modulus of continuity, local regularity, dilation matrix,
transition matrix, joint spectral radius, invariant polytope algorithm

Classification (MSCS): 65D17, 15A60, 39A99

Dedicated to the memory of Maria Charina

##  1 Introduction

1.1. Refinement equations with matrix dilation

Refinement equations are difference functional equations with an integer
contraction of the argument. They play a crucial role in the construction of
wavelets in $L_{2}({\mathbb{R}}^{n})$ and of the subdivision algorithms in the
approximation theory and geometrical design, see surveys in [2, 4, 37, 38].
They are also applied to random power series [16, 28, 40], combinatorial
number theory [19, 42], discrete geometry (tilings) [20, 32], etc. We consider
the multivariate equation of the form:

$\varphi({\boldsymbol{x}})\quad=\quad\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}\,c_{{\boldsymbol{k}}}\,\varphi(M{\boldsymbol{x}}-{\boldsymbol{k}})\,,\qquad{\boldsymbol{x}}\in{\mathbb{R}}^{n},$
(1)

where the dilation matrix $M$ is integer and expanding, i.e., has integer
entries and all eigenvalues bigger than one by modulo. We restrict ourselves
to equations with a finite summation, i.e., $c_{{\boldsymbol{k}}}=0$ if
${\boldsymbol{k}}\notin Q$, where $Q$ is a finite nonempty subset of
${\mathbb{Z}}^{n}$. We also make the standard assumption
$\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}c_{{\boldsymbol{k}}}\,=\,m,$ where
$m=|\det M|$. In this case, equation (1) possesses a unique, up to
multiplication by a constant, solution $\varphi$ (refinable function) in the
space ${\mathcal{S}}_{0}^{\prime}({\mathbb{R}}^{n})$ of compactly-supported
tempered distributions. In most of applications this solution needs to be at
least from $L_{2}({\mathbb{R}}^{n})$. Often a higher smoothness is required.
Let us note that $\varphi$ is never infinitely smooth (in [50] this was proved
for more general equations).

In the univariate case ($n=1$) the smoothness of refinable functions has been
studied in great detail. Two methods are usually most prominent. The matrix
approach [6, 5, 15, 36, 41] finds the biggest $k$ such that $\varphi\in
C^{k}({\mathbb{R}})$ and computes the precise value of the Hölder exponent in
this space. This method requires computation of the joint spectral radius of
special matrices $\\{T_{i}\\}_{i=0}^{m-1}$, which is a hard problem even in
low dimensions. Recent tools elaborated in [21, 22, 34] can do this for
matrices of non very large sizes, which makes it possible to compute the
regularity for univariate equation
$\varphi(x)=\sum_{k=0}^{N}\,c_{k}\varphi(2x-k)$ of order $N\leq 25$. The
matrix method is also applicable to compute the regularity in
$L_{2}({\mathbb{R}})$ [33] and in the Sobolev spaces $W_{2}^{k}({\mathbb{R}})$
(of absolutely continuous functions $f$ such that $f^{(k)}\in L_{2}$). In this
case the joint spectral radius computation can be replaced by computing the
usual spectral radius of one matrix $T$, although, of quite a big size, around
$\frac{1}{2}\,N^{2}$, see [39].

The second approach is usually referred to as the Sobolev regularity
computation or the Littlewood-Paley type method. It finds the regularity in
$W_{2}^{k}({\mathbb{R}})$ by means of computing the Sobolev regularity defined
as $\sup\,\bigl{\\{}\beta\geq
0\,:\,\int_{{\mathbb{R}}}|\varphi(\xi)|^{2}(1+|\xi|^{2})^{\beta}\,dt\,<\,\infty\bigr{\\}}$.
This problem is reduced to finding the usual spectral radius of a matrix of a
much lower size, around $2N$ [13, 17, 18, 49]. The shortcoming of this method
is that it deals with the regularity not in $C$ but in $L_{2}$. Say, to prove
the continuity of $\varphi$, one needs to show that $\varphi\in W^{1}_{2}$.
The matrix method and the Littlewood-Paley type method are both widely used.

For multivariate refinement equations, the situation is more complicated. The
smoothness of solutions has been computed only in the special case when the
matrix $M$ is isotropic, i.e., is similar to a diagonal matrix with equal by
modulus eigenvalues. Isotropic matrices expand all distances equally in all
directions, i.e., the sequence
$\\{M^{k}{\boldsymbol{x}}\\}_{k\in{\mathbb{N}}}$ has the same asymptotic
growth for all ${\boldsymbol{x}}\in{\mathbb{R}}^{n}\setminus\\{0\\}$. In this
case many results on the univariate equations can be generalized to
${\mathbb{R}}^{n}$. In particular, both of the aforementioned methods work for
isotropic dilations, although the dimensions of the corresponding matrices
$\\{T_{i}\\}_{i=1}^{m-1}$ and $T$ become much higher. This was done in [4, 6,
12, 13, 23, 24, 25, 26, 27, 30, 45]. If $M$ is not isotropic, then none of the
methods is applicable. Already simple two-dimensional examples show that the
formulas for the Hölder exponent are not valid in the non-isotropic case. Even
some basic questions, for instance, whether or not $\varphi$ possesses a
derivative (say, from $C({\mathbb{R}}^{n})$ or from
$L_{2}({\mathbb{R}}^{n})$), remain unanswered. Here it should be emphasized
that refinement equations with non-isotropic dilations are important in many
problems of signal processing, wavelets and geometric design, see [1, 2, 8, 9,
10, 11, 30]. In [2, 7, 14, 47] the authors compute regularity in special
anisotropic Hölder and Sobolev spaces.

It was not before 2019 that the modification of the matrix method was
elaborated for non-isotropic dilations [11]. This gave closed formulas for the
Hölder exponent and for higher order regularity of general refinable functions
by means of joint spectral radii. However, because of the large size of the
matrices $T_{i}$, this method is applicable mostly for very low dimensions $n$
and for small sets $Q$ of nonzero coefficients $c_{{\boldsymbol{k}}}$.

An improvement would be achieved if one could make the next step: to modify
the Littlewood-Paley type method for non-isotropic matrices. In this paper we
make this step. We obtain a sharp criterion which, for an arbitrary dilation
matrix $M$, ensures that the solution of refinement equation (1) belongs to
$W^{k}_{2}({\mathbb{R}}^{n})$ and presents a formula for its Hölder exponent.
This formula is different and more complicated than one for the isotropic
case, although, for many equations, it can be simplified. Which is most
important, it allows us to solve the smoothness problem in higher dimensions
$n$ and for larger sets $Q$. Numerical results in Section 7 demonstrate that
indeed the Littlewood-Paley method for computing the $L_{2}$-regularity of
multivariate refinement equations is more efficient than the matrix approach
because of a much smaller dimension of the problem of the spectral radius
computation.

Then we apply this formula to the tile B-splines and to the corresponding
subdivision schemes.

1.2. The tile B-splines

In the second part of the paper we analyse the regularity of the tile
B-splines recently introduced in [51]. They are analogues of the univariate
cardinal B-splines. The cardinal B-spline of order $\ell\geq 0$ is
$B_{\ell}\,=\,\underbrace{\chi_{[0,1]}*\cdots*\chi_{[0,1]}}_{\ell+1}$, it has
integer nodes, is supported on $[0,\ell+1]$, and belongs to
$W_{2}^{\ell}({\mathbb{R}}^{n})$. In particular $B_{0}=\chi_{[0,1]}$. Cardinal
B-splines have countless applications due to their simplicity and many optimal
properties (the smoothest refinable function of a given order, etc.) The
multivariate B-splines are defined in the standard way as direct products of
univariate ones. This approach, however, has significant shortcomings,
especially in the construction of wavelets and Haar systems, see [5, 7, 30]
for general discussion of this aspect. In contrast, the tile B-splines are not
obtained by direct products but by the convolutions:
$\varphi_{\ell}\,=\,\chi_{G}*\cdots*\chi_{G}$, where $G$ is a self-similar
tile in ${\mathbb{R}}^{n}$. Here we need to recall some definitions.

Given an integer expanding $n\times n$ matrix $M$, we take a system of
“digits” $D=\\{{\boldsymbol{d}}_{0},\ldots,{\boldsymbol{d}}_{m-1}\\}$, which
are $m=|\det M|$ arbitrary representatives of the quotient group
${\mathbb{Z}}^{n}/M{\mathbb{Z}}^{n}$. The set $G$ of $M$-adic fractions:

$G\ =\ \Bigl{\\{}0.{\boldsymbol{d}}_{i_{1}}{\boldsymbol{d}}_{i_{2}}\cdots\ =\
\sum_{k\in{\mathbb{N}}}\,M^{-k}{\boldsymbol{d}}_{i_{k}}\ :\
{\boldsymbol{d}}_{i_{k}}\in D\,,\ k\in{\mathbb{N}}\Bigr{\\}}$

is a self-similar attractor. This is a compact set of an integer positive
Lebesgue measure [20, 32]. If its measure is equal to one, then the attractor
is called a tile. Integer shifts of a tile form a partition of
${\mathbb{R}}^{n}$. Moreover, a tile is self-similar: it is a union of several
parallel shifts of the set $M^{-1}G$. Its indicator $\varphi_{0}=\chi_{G}$
satisfies the refinement equation
$\varphi_{0}({\boldsymbol{x}})\,=\,\sum_{i=0}^{m-1}\varphi_{0}(M{\boldsymbol{x}}-{\boldsymbol{d}}_{i})$.
In some sense, the tile is a unit segment in the $M$-adic system on
${\mathbb{R}}^{n}$. The Haar systems in $L_{2}({\mathbb{R}}^{n})$ with the
dilation factor $M$ can be obtained from the function $\chi_{G}$ by the same
procedure as the classical Haar system with the double dilation in
$L_{2}({\mathbb{R}})$ is obtained from $\chi_{[0,1]}$. To construct wavelet
bases in $L_{2}({\mathbb{R}}^{n})$ one can naturally define the tile B-spline
$\varphi_{\ell}$ as a convolution of $\ell+1$ functions $\varphi_{0}$.
Strictly speaking, $\varphi_{\ell}$ is not a spline, since it is not a
piecewise-polynomial function. On the other hand, $\varphi_{\ell}$ is smooth,
generates algebraic polynomials of degrees $\ \leq\ell$ by its integer
translates, and the values $\varphi_{\ell}({\boldsymbol{x}})$ are easily
computed in all $M$-adic points ${\boldsymbol{x}}$ (from the corresponding
refinement equation). The tile B-splines are useful in approximations and
interpolations as well as for constructing of multivariate wavelets and
subdivision schemes [51].

Applying our main results, we prove that some families of tile B-splines have
(quite surprisingly) a higher smoothness than the standard B-splines of the
same order. Note that nothing of this kind occurs in the univariate case,
where the cardinal B-spline is the smoothest refinable function of a given
order. This “super-smoothness” yields, in particular, that the subdivision
schemes defined by tile B-splines generate smoother surfaces and possess
faster convergence. Moreover, in Section 8 we prove the optimality of tile
B-splines among all refinable functions of a given order.

1.3. The tile subdivision schemes

The tile B-splines demonstrate their efficiency in the construction of
multivariate wavelets [51] and of subdivision schemes. The subdivision
operator
$S:\,\ell_{\infty}({\mathbb{Z}}^{n})\to\ell_{\infty}({\mathbb{Z}}^{n})$
generated by a dilation matrix $M$ and coefficients
$\\{c_{{\boldsymbol{k}}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$ is defined
as
$\,[S{\boldsymbol{a}}](\cdot)\,=\,\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}c_{\cdot-M{\boldsymbol{k}}}\,{\boldsymbol{a}}({\boldsymbol{k}})$.
The subdivision scheme is a recursive application of $S$ to a function
$f_{0}:{\mathbb{Z}}^{n}\,\to{\mathbb{R}}$. The subdivision scheme converges to
a limit function $f\in C({\mathbb{R}}^{n})$ if
$\bigl{\|}f(M^{-j}\cdot)\,-\,S^{j}f_{0}(\cdot)\bigr{\|}_{\infty}\to\infty$ as
$j\to\infty$. Similarly one defines the convergence in $L_{2}$. In Section 9
we analyse the properties of subdivision schemes generated by the tile
B-splines, in particular, those having the super-smoothness. We prove that the
tile schemes are optimal in the following sense: they have the lowest possible
complexity of one iteration among all schemes of a given approximation rate.

1.4. Notation

We denote vectors by bold letters and scalars by usual ones, so
${\boldsymbol{x}}\,=\,(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$. The
polynomials are also denoted by bold letters. The cardinality of a finite set
$X$ is denoted by $|X|$.

We define the Fourier transform in ${\mathbb{R}}^{n}$ by the formula
$\widehat{f}({\boldsymbol{\xi}})\,=\,\int_{{\mathbb{R}}^{n}}f({\boldsymbol{x}})e^{-2\pi
i({\boldsymbol{x}},{\boldsymbol{\xi}})}d{\boldsymbol{x}}$. As usual, the
convolution of functions $f,g$ is defined by $f*g(x)\ =\ \int f(t)g(x-t)dt$,
also we use the symmetrized autoconvolution of a function $\varphi$ defined as
$\varphi*(\varphi\\_)$, where
$\varphi\\_(\mathord{\,\cdot\,})=\varphi(-\mathord{\,\cdot\,})$.

We consider trigonometric polynomials of the form
${\boldsymbol{p}}({\boldsymbol{\xi}})\,=\,\sum_{{\boldsymbol{k}}\in\Omega}c_{{\boldsymbol{k}}}e^{-2\pi
i({\boldsymbol{k}},\xi)},\,{\boldsymbol{\xi}}\in{\mathbb{R}}^{n}$, where
$\Omega$ is a finite subset of ${\mathbb{Z}}^{n}$. We usually drop the word
“trigonometric”. The set of zeros of a function $f$ will be denoted as
$\mbox{\em nill}\,(f)\,=\,\\{{\boldsymbol{x}}\in{\mathbb{R}}^{n}:\
f({\boldsymbol{x}})=0\\}$. Similarly, for a natural number $\ell$, $\mbox{\em
nill}\,_{\ell}(f)$ denotes the set of zeros of order at least $\ell+1$. For
any closed subset $C\subset{\mathbb{R}}^{n}$, let $\pi\\{C\\}$ be the set of
points ${\boldsymbol{\xi}}\in{\mathbb{R}}^{n}$ such that
${\boldsymbol{\xi}}+{\boldsymbol{k}}\in C$ for all
${\boldsymbol{k}}\in{\mathbb{Z}}^{n}$. For several subsets $C_{i}$, we write
$\pi\,\\{C_{1},\ldots,C_{m}\\}$ for $\pi\\{C_{1}\cup\ldots\cup C_{m}\\}$.

Two functions $f(x),g(x)$ on ${\mathbb{R}}$, are asymptotically equivalent
$\,\bigl{(}f(x)\ {\asymp}\ g(x)\bigr{)}$ if there exist positive $c_{1},c_{2}$
such that $\,c_{1}|f(x)|\,\leq\,|g(x)|\,\leq\,c_{2}|f(x)|,\ x\in{\mathbb{R}}$.
We also define the exponential asymptotic equivalence. For a given sequence
$\\{a_{k}\\}_{k=1}^{\infty}$ and $\lambda>0$, we write $a_{k}\ \dot{\asymp}\
\lambda^{k}$ if there exist positive $r,c_{1},c_{2}$ such that

$c_{1}\lambda^{k}\ \leq\ |a_{k}|\ \leq\ c_{2}k^{r}\lambda^{k},\
k\in{\mathbb{N}}.$

We write $|a_{k}|\ \dot{\leq}\ \lambda^{k}$ if there exist positive constants
$r,c$ such that $|a_{k}|\ \leq\ c\,k^{r}\,\lambda^{k},\,k\in{\mathbb{N}}$.

For a given subset $X\subset{\mathbb{R}}^{n}$, we denote by ${\rm
cone}\,X\,=\,{\rm co}\,\\{\lambda{\boldsymbol{x}}\ :\ {\boldsymbol{x}}\in
X,\,\lambda\in{\mathbb{R}}_{+}\\}$ the conic hull of $X$.

The function
${\boldsymbol{c}}({\boldsymbol{\xi}})\,=\,\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}c_{{\boldsymbol{k}}}e^{-2\pi
i({\boldsymbol{k}},{\boldsymbol{\xi}})}$ is the mask of the refinement
equation (1). We say that the mask is supported on a set
$\Omega\subset{\mathbb{Z}}^{n}$ if $c_{{\boldsymbol{k}}}=0$ for
${\boldsymbol{k}}\notin\Omega$. Since we assume that the support is finite,
${\boldsymbol{c}}({\boldsymbol{\xi}})$ is a trigonometric polynomial.

##  2 The roadmap for the main results

2.1. Computing the regularity in the isotropic case: an overview

The matrix approach makes use of the transition operator:

$[{\mathcal{F}}f](\mathord{\,\cdot\,})\quad=\quad\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}c_{{\boldsymbol{k}}}f(M\mathord{\,\cdot\,}\,-\,{\boldsymbol{k}}),$

where, recall, the summation is finite and the mask is supported on the subset
$\Omega\subset{\mathbb{Z}}^{n}$. The refinable function $\varphi$ is the
eigenfunction of ${\mathcal{F}}$ with the eigenvalue one. Assume that $M$ is
isotropic (including the case $n=1$), $r=\rho(M)$ is the spectral radius of
$M$, $W$ is the linear space spanned by the differences
$\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,}),\,{\boldsymbol{h}}\in{\mathbb{R}}^{n}$.
From the refinement equation it follows that the value
$\sup_{\|{\boldsymbol{h}}\|\leq
r^{-k}}\|\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})\|_{\infty}$
is equivalent to $\|{\mathcal{F}}^{k}|_{W}\|$. This implies that the Hölder
exponent of $\varphi$ is equal to $\log_{\,1/r}\,\rho({\mathcal{F}}|_{W})$ [6,
4]. Let us stress that this is true only if $M$ is isotropic. It is proved
that the spectral radius $\rho({\mathcal{F}}|_{W})$ in the space
$C({\mathbb{R}}^{n})$ is equal to the so-called joint spectral radius of
$N\times N$ transition matrices $\\{T_{0},\ldots,T_{m-1}\\}$ constructed by
the coefficients $c_{{\boldsymbol{k}}}$ and restricted to a certain subspace
of ${\mathbb{R}}^{N}$. The computation of the joint spectral radius, however,
is a hard problem. The Hölder exponent in $L_{2}$ is expressed by the
$L_{2}$-radius of the same matrices. It is equal to the square root of the
usual spectral radius of the matrix
$T=\frac{1}{m}\,\sum_{i=0}^{m-1}\,T_{i}\otimes T_{i}$, where $\otimes$ denotes
the Kronecker product [39]. Thus, the computation of the $L_{2}$-regularity by
the matrix method involves the usual spectral radius $\rho(T)$. However, $T$
has the size of order $\frac{1}{2}\,N^{2}$ while the matrices $T_{i}$ have
size of order $N$.

We consider the Littlewood-Paley type method (computation of Sobolev
regularity). Its idea is to associate to every function $f\in W$ the
periodization of the square of its Fourier transform $\widehat{f}$:

$\Phi_{f}\,({\boldsymbol{\xi}})\quad=\quad\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}\bigl{|}\widehat{f}({\boldsymbol{\xi}}+{\boldsymbol{k}})\bigr{|}^{2}\,.$
(2)

The Plancherel equality yields that
$\|\Phi_{f}\|_{1}\,=\,\|\widehat{f}\|_{2}^{2}\,=\,\|f\|_{2}^{2}$ (the norm of
$\Phi_{f}$ is computed in $L_{1}({\mathbb{T}}_{n})$). If we apply the operator
${\mathcal{F}}$ to the function $f$, what happens then to $\Phi_{f}$ ? To
answer this question, we define the operator ${\mathcal{T}}$ on
$L_{2}({\mathbb{T}}_{n})$ as follows. Let
$D^{*}\,=\,\\{{\boldsymbol{d}}_{i}^{*}\\}_{i=0}^{m-1}$ be a set of digits for
the transpose matrix $M^{T}$. This set may not coincide with $D$. Then

$\bigl{[}{\mathcal{T}}{\boldsymbol{p}}\bigr{]}({\boldsymbol{\xi}})\quad=\quad\sum_{j=0}^{m-1}\,\bigl{|}\,{\boldsymbol{c}}\,(M^{-T}({\boldsymbol{\xi}}+{\boldsymbol{d}}^{*}_{j}))\bigr{|}^{2}\,{\boldsymbol{p}}\bigl{(}\,M^{-T}({\boldsymbol{\xi}}+{\boldsymbol{d}}^{*}_{j})\,\bigr{)}\
,\qquad{\boldsymbol{p}}\in L_{2}({\mathbb{T}}_{n})\,.$ (3)

A direct calculation shows that
$\,\Phi_{\,{\mathcal{F}}f}\,=\,{\mathcal{T}}\,[\Phi_{f}]$. Hence, for each
$k\geq 1$, the norm $\|{\mathcal{F}}^{k}f\|_{2}$, is equal to the square root
of the $L_{1}$-norm of ${\mathcal{T}}^{k}[\Phi_{f}]$. One may wonder what is
the use of changing one infinite-dimensional operator ${\mathcal{F}}$ to
another operator ${\mathcal{T}}$? Observe that, for every compactly supported
$f\in W$, the function $\Phi_{f}$ is a polynomial. Indeed, $\Phi_{f}$ has the
following Fourier expansion:

$\Phi_{f}({\boldsymbol{\xi}})\ =\
\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}\,\Bigl{(}\,f(\mathord{\,\cdot\,})\,,\,f(\mathord{\,\cdot\,}+{\boldsymbol{k}})\,\Bigr{)}\,e^{\,2\pi
i\,({\boldsymbol{k}},{\boldsymbol{\xi}})}\,,$

where only finitely many coefficients
$\bigl{(}f(\cdot),f(\cdot+{\boldsymbol{k}})\bigr{)}$ are nonzero. This
argument reduces, the operator ${\mathcal{T}}$ to a finite-dimensional space
of polynomials ${\mathcal{P}}_{\Omega}$ with coefficients supported on certain
finite symmetric subset $\Omega\subset{\mathbb{Z}}^{n}$. It is possible to
prove that if $f\in W$, then the polynomial $\Phi_{f}$ has a root of
multiplicity two at zero. Denoting the space of such polynomials by
${\mathcal{P}}_{0}={\mathcal{P}}_{0}(\Omega)$, we have
$\rho^{2}({\mathcal{F}}|_{W})\,=\,\rho({\mathcal{T}}|_{{\mathcal{P}}_{0}})$,
and therefore, the Hölder exponent of $\varphi$ in $L_{2}$ is equal to
$\log_{1/r}\,\rho({\mathcal{F}}|_{W})\,=\,\frac{1}{2}\,\log_{1/r}\rho({\mathcal{T}}|_{{\mathcal{P}}_{0}})$.
Note that the dimension of the space ${\mathcal{P}}_{\Omega}$ is usually much
less than the size of the matrix $T$, which makes this formula more efficient
in practice and applicable for a wider set of refinement equations. This
holds, however, only for the isotropic case. The situation in the case of
general matrix $M$ is quite different.

2.2. What is wrong with the non-isotropic case?

The advantage of the Littlewood-Paley type method is that it expresses the
exponent of Sobolev and of $L_{2}$ regularity merely by the spectral radius of
one matrix, whose size is usually not too large. Neither the joint spectral
radius nor the Kronecker lifting is involved. However, the formula
$\,\frac{1}{2}\log_{1/r}\,\rho({\mathcal{T}}|_{{\mathcal{P}}_{0}})$ is valid
only for isotropic $M$. For general $M$, neither this formula, nor its
possible modifications (for example, by replacing $r$ with
$|{\rm\det}\,M|^{1/n}$) holds. The reason is that a non-isotropic matrix $M$
expands the distances differently in different directions. This aspect has to
be taken into account in any regularity analysis of non-isotropic refinement
equations. Can the Littlewood-Paley type method be applied to arbitrary
dilation matrices?

2.3. Our solution for the non-isotropic case: the four steps

The idea of handling the non-isotropic case was suggested in [11] and worked
out in the space $C({\mathbb{R}}^{n})$. One considers the spectral subspaces
$J_{1},\ldots,J_{q}$ of the matrix $M$. Each subspace $J_{s}$ corresponds to
all eigenvalues of $M$ of a given modulus $r_{s}$. Thus, $r_{1}>\cdots>r_{q}$
are all moduli of eigenvalues of $M$ and the restriction $M|_{J_{s}}$ has all
the eigenvalues of modulus $r_{s}$. Note that the operator $M|_{J_{s}}$ is not
necessarily isotropic since it may have nontrivial Jordan blocks. We have
${\mathbb{R}}^{n}\,=\,J_{1}\oplus\cdots\oplus J_{q}$ (the direct sum). The
regularity of functions on ${\mathbb{R}}^{n}$ is considered separately on the
directions of the subspaces $J_{s}$. This is done by computing the spectral
radii of the restrictions of the operator ${\mathcal{F}}$ to the subspaces
$W_{s}$ spanned by the differences
$\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,}),\,{\boldsymbol{h}}\in
J_{s}$. Then the regularity on the whole ${\mathbb{R}}^{n}$ is the minimal
regularity along $J_{s},\,s=1,\ldots,q$. This idea works for the matrix method
[11], however, its extension to the Littlewood-Paley type method meets
considerable difficulties. One of the reasons is that the map:
$\Phi:\,W\to{\mathcal{P}}_{0}$, which associates $\Phi_{f}$ to a function
$f\in W$, is nonlinear. Hence, the images of the subspaces $W_{s}$ become
nonlinear manifolds in ${\mathcal{P}}_{0}$. We handle this problem by a new
construction realized in four steps described below.

Step 1. We use the spectral properties of linear operators with an invariant
cone. Theorem 3.3, the main result of Section 3, states that if
$K\subset{\mathbb{R}}^{n}$ is a closed solid pointed cone respected by a
linear operator $A$, then for every point ${\boldsymbol{x}}\in K$, the
asymptotic growth of the trajectory
$\\{A^{j}{\boldsymbol{x}}\\}_{j=1}^{\infty}$ as $j\to\infty$ is determined by
the spectral radius of $A$ restricted to the minimal face of $K$ invariant
with respect to $A$ and containing ${\boldsymbol{x}}$. The same is true not
only for one point ${\boldsymbol{x}}$ but also for an arbitrary subset
$\Gamma\subset K$. The fastest asymptotic growth of trajectories
$\\{A^{j}{\boldsymbol{x}}\\}_{j=1}^{\infty}$ over all points
${\boldsymbol{x}}\in\Gamma$ is defined by the minimal invariant face of $K$
containing $\Gamma$.

Step 2. Theorem 3.3 is applied for the cone
${\mathcal{P}}_{+}={\mathcal{P}}_{+}(\Omega)$ of nonnegative trigonometric
polynomials
${\boldsymbol{p}}({\boldsymbol{x}})\,=\,\sum_{{\boldsymbol{k}}\in\Omega}\,p_{{\boldsymbol{k}}}\,e^{2\pi
i\,({\boldsymbol{k}},{\boldsymbol{x}})}$. To this end, we characterize all
invariant faces of the cone ${\mathcal{P}}_{+}$. Propositions 4.1 and 4.2
assert that, for an arbitrary operator $A$ leaving ${\mathcal{P}}_{+}$
invariant, every invariant face $L\subset{\mathcal{P}}_{+}$ is defined by a
set $S\subset{\mathbb{R}}^{n}$ of common zeros, i.e., $\
L=\bigl{\\{}{\boldsymbol{p}}\in{\mathcal{P}}_{+}\ :\
{\boldsymbol{p}}(S)=0\bigr{\\}}$.

Step 3. We consider an arbitrary difference
$f=\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})$
along a shift vector ${\boldsymbol{h}}\in J_{s}$. Applying contractions by the
powers of $M^{-1}$, we obtain a small vector $M^{-j}{\boldsymbol{h}}$ of the
norm equivalent to $r_{s}^{-j}$. On the other hand, the corresponding function
$\Phi_{f}$ after changing the shift vector from ${\boldsymbol{h}}$ to
$M^{-j}{\boldsymbol{h}}$, becomes ${\mathcal{T}}^{j}\Phi_{f}$, where the
linear operator ${\mathcal{T}}$ is given by (3). Now we apply the results of
Steps 1, 2 to the positive polynomial cone ${\mathcal{P}}_{+}$ and conclude
that $\|{\mathcal{T}}^{j}\Phi_{f}\|\ \dot{\asymp}\ (\rho_{s})^{j}$, where
$\rho_{s}$ is the spectral radius of ${\mathcal{T}}$ restricted to a certain
face $K_{s}$ of ${\mathcal{P}}_{+}$. Thus,
$\|\varphi(\mathord{\,\cdot\,}+M^{-j}{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})\|^{2}\,=\,\
\|{\mathcal{T}}^{j}\Phi_{f}\|_{1}\ \dot{\asymp}\ (\rho_{s})^{j}$. We see that
while the norm of the shift ${\boldsymbol{h}}\in J_{s}$ is multiplied by
(roughly) $r_{s}^{-j}$, the norm of the variation of $\varphi$ is multiplied
by $(\rho_{s})^{j}$. Now it is easily deduced that the Hölder exponent of
$\varphi$ along the subspace $J_{s}$ is equal to $\,\log_{1/r_{s}}\rho_{s}$.
The minimum of those values over $s=1,\ldots,q$ gives the Hölder exponent of
$\varphi$.

Note that if the dilation matrix $M$ is isotropic, then there is only one
spectral subspace $J_{1}={\mathbb{R}}^{n}$ and hence $W_{1}=W$. Therefore,
$\Phi$ maps $W_{1}$ to the entire space ${\mathcal{P}}_{0}$, and the Hölder
exponent of $\varphi$ is expressed merely by the spectral radius of
${\mathcal{T}}|_{{\mathcal{P}}_{0}}$. This was the core of the method in the
isotropic case [13, 18, 25, 27]. Now, dealing with an arbitrary matrix $M$, we
may have proper subspaces $W_{1},\ldots,W_{q}$ of ${\mathbb{R}}^{n}$. The
image of $W_{s}$ by the map $\Phi$ is some nonlinear manifold in
${\mathcal{P}}_{0}$. If one follows the line of reasoning from the isotropic
case, he needs to apply the operators ${\mathcal{T}}^{j}$ to those manifolds,
which can be difficult. We resolve this problem by means of invariant cones.
Steps 1 and 2 allow us to to apply the operator ${\mathcal{T}}^{j}$ to the
minimal invariant face $K_{s}$ of the cone ${\mathcal{P}}_{+}$ containing
those manifolds. This minimal face $K_{s}$ can be found explicitly, which
leads to the formula for the Hölder exponent in $L_{2}$ (Theorem 5.3).

Step 4. To extend our approach to the higher order regularity we use the $k$th
order differences of the function $\varphi$ along the shift vectors
${\boldsymbol{h}}_{i}\in{\mathbb{R}}^{n}$. Replacing the difference
$\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})$
by the higher order difference, we adopt our approach to the higher order
regularity and obtain an explicit formula for the Hölder exponents of the
derivatives in Theorem 5.7.

2.4. Applications to tiles, B-splines, and subdivisions

We use our method (Theorems 5.3 and 5.7) to find the regularity of the tile
B-splines. To this end, one needs to find the invariant faces $K_{s}$ of the
cone ${\mathcal{P}}_{+}$. This is done in Section 6 by involving the geometry
of cylinders in ${\mathbb{R}}^{n}$ composed by tiles and some known facts on
the rational subspaces of ${\mathbb{R}}^{n}$ (Theorem 6.4). We shall see that
the faces $K_{s}$ in case of tile B-splines have a simple structure, which
leads to the precise values of the high order Hölder exponents (Theorem 6.8).
In particular, we find several types of the tile B-splines that have the
regularity higher than the traditional cardinal B-splines and the faster
convergence of the corresponding subdivision schemes. The efficiency of the
tile subdivisions is demonstrated in Section 9 along with other examples and
numerical results.

##  3 Auxiliary results. Operators leaving a cone invariant

We consider a convex cone $K\subset{\mathbb{R}}^{d}$, which is a set closed
with respect to addition and multiplication by positive numbers: if
${\boldsymbol{a}},{\boldsymbol{b}}\in K$, then
$\,{\boldsymbol{a}}+{\boldsymbol{b}}\in K$ and $\lambda{\boldsymbol{a}}\in K$
for every $\lambda\in{\mathbb{R}}_{+}$. In what follows we always assume that
a cone is closed, solid (or full-dimensional, i.e., possesses a nonempty
interior) and pointed (does not contain a straight line). A linear operator
$A$ in ${\mathbb{R}}^{d}$ leaves $K$ invariant if $\ AK\subset K$. By the
Krein-Rutman theorem [29], an operator leaving a cone $K$ invariant possesses
a leading eigenvector ${\boldsymbol{v}}\in K,\,{\boldsymbol{v}}\neq 0$, for
which $A{\boldsymbol{v}}=\lambda_{\max}{\boldsymbol{v}},\
\lambda_{\max}=\rho(A)$. This fact extends the well-known Perron-Frobenius
theorem from the positive orthant ${\mathbb{R}}^{d}_{+}$ to an arbitrary cone
$K\subset{\mathbb{R}}^{d}$.

A plane of support of a cone $K$ is a hyperplane that intersects $K$ but does
not intersect its interior. A face of a cone is its intersection with a plane
of support. In particular, the apex ${\boldsymbol{0}}$ is the face of
dimension zero. In addition, the cone $K$ itself is considered as the face of
dimension $d$. All other faces are proper and have dimensions from one (the
edges) to $d-1$ (the facets). All proper faces are cones of smaller
dimensions. A face plane is a linear span of a face. For an arbitrary subset
$\Gamma\subset K$, its minimal face is the smallest by inclusion face of $K$
that contains $\Gamma$. If an operator $A$ leaves $K$ invariant, then the
minimal invariant face for $\Gamma$ is the smallest by inclusion face of $K$
invariant with respect to $A$ and containing $\Gamma$.

We are interested in the asymprotic behaviour of the trajectory
$A^{j}{\boldsymbol{x}}$ as $j\to\infty$, for a given point ${\boldsymbol{x}}$.
Clearly, $\|A^{j}{\boldsymbol{x}}\|\,\dot{\leq}\,\rho^{\,j}$, where
$\rho=\rho(A)$ is the spectral radius of $A$. If ${\boldsymbol{x}}$ belongs to
an invariant cone, one can say more:

###### Proposition 3.1

Let an operator $A$ leave a cone $K$ invariant and let $\rho=\rho(A)$. Then
there exists a face $\bar{L}$ of $K$ invariant with respect to $A$, such that
${\rm dim}\,\bar{L}\leq d-1$ and for every ${\boldsymbol{x}}\in
K\setminus\bar{L}$, we have
$\|A^{j}{\boldsymbol{x}}\|\,\dot{\asymp}\,\rho^{\,j},\ j\in{\mathbb{N}}$.

Proof. If all eigenvalues of $A$ have equal absolute values $\rho$, then the
assertion is obvious true with $\bar{L}=\\{0\\}$. Otherwise, denote by
$\rho_{2}$ the second largest modulus of eigenvalues of $A$. Thus,
$\rho_{2}<\rho$ and all vectors ${\boldsymbol{x}}$ such that
$\|A^{j}{\boldsymbol{x}}\|\,\dot{\leq}\,\rho_{2}^{\,j}$ form a proper
invariant subspace $V$ of the operator $A$. For every
${\boldsymbol{x}}\in{\mathbb{R}}^{d}$, we have
$\|A^{j}{\boldsymbol{x}}\|\,\dot{\asymp}\,\rho^{\,j}$, unless
${\boldsymbol{x}}\in V$. Denote $K^{\prime}=K\setminus\\{{\boldsymbol{0}}\\}$.
If $V$ does not intersect $K^{\prime}$, then the proposition follows with
$\bar{L}=\\{{\boldsymbol{0}}\\}$. If it intersects the interior of $K$, then
we take an arbitrary point ${\boldsymbol{z}}\in{\rm int}\,K\,\cap\,V$ and
consider the vector ${\boldsymbol{z}}-t{\boldsymbol{v}}$, where
${\boldsymbol{v}}$ is the Krein-Rutman eigenvector, for which
$A{\boldsymbol{v}}=\rho\,{\boldsymbol{v}}$, and $t>0$ is small enough, so
${\boldsymbol{z}}-t{\boldsymbol{v}}\in K$. We have
$A^{j}({\boldsymbol{z}}-t{\boldsymbol{v}})\in K$ for all $j$. On the other
hand,
$A^{j}({\boldsymbol{z}}-t{\boldsymbol{v}})\,=\,A^{j}{\boldsymbol{z}}\,-\,tA^{j}{\boldsymbol{v}}\,=\,A^{j}{\boldsymbol{z}}\,-\,t\,\rho^{\,j}{\boldsymbol{v}}\,=\,-t\,\rho^{\,j}{\boldsymbol{v}}\,+\,o(\rho^{j})$
as $j\to+\infty$. However,
$-t\,\rho^{\,j}{\boldsymbol{v}}\,+\,o(\rho^{j})\,=\,\rho^{\,j}\bigl{(}-t{\boldsymbol{v}}\,+\,o(1)\bigr{)}\notin
K$ for large $j\to+\infty$, which is a contradiction. Thus, $V$ cannot
intersect the interior of $K$. It remains to consider the last case: $V$
intersects $K^{\prime}$ but $V\cap{\rm int}\,K=\emptyset$. By the convex
separation theorem, $V$ can be separated from $K$ by a hyperplane $H$. The set
$H\cap K$ is a proper face of $K$ that contains the set $V\cap K$. Since the
operator $A$ respects the set $V\cap K$, it respects its minimal face, which
will be denoted by $\bar{L}$. Thus, $\bar{L}$ is an invariant face, every
point ${\boldsymbol{x}}\in K\setminus\bar{L}$ does not lie in $V$, and hence
$\|A^{j}{\boldsymbol{x}}\|\,\dot{\asymp}\,\rho^{\,j}$.

$\Box$

###### Corollary 3.2

Suppose a linear operator $A$ leaves a cone $K\subset{\mathbb{R}}^{d}$
invariant; then, for every ${\boldsymbol{x}}\in K$, we have

$\bigl{\|}A^{j}{\boldsymbol{x}}\bigr{\|}\quad\asymp\quad\bigl{\|}A^{j}\bigl{|}_{L}\bigr{\|}\quad\dot{\asymp}\quad\rho^{\,j}\bigl{(}A\bigl{|}_{L}\bigr{)},\qquad
j\in{\mathbb{Z}},$ (4)

where $L$ is the minimal invariant face of ${\boldsymbol{x}}$.

Proof. It suffices to consider the case $\|{\boldsymbol{x}}\|=1$ and $L=K$,
otherwise we restrict $A$ to the corresponding face plane. If $\rho(A)=0$,
then $A$ is a nilpotent operator, $A^{j}=0$ for $j\geq d$, and (4) holds. If
$\rho(A)>0$, then by Proposition 3.1, the assertion (4) holds for all
${\boldsymbol{x}}\in K$ which do not belong to a special invariant face of
$K$. However, by the assumption, ${\boldsymbol{x}}$ does not belong to proper
invariant faces, which concludes the proof.

$\Box$

We are ready to prove the main result of this section, which gives us the key
to computing the regularity of refinable functions.

###### Theorem 3.3

Let a linear operator $A$ leave a cone $K\subset{\mathbb{R}}^{d}$ invariant
and $\Gamma$ be an arbitrary subset of $K$. Let also $L$ be the minimal
invariant face for $\Gamma$. Then for every ${\boldsymbol{x}}\in\Gamma$, we
have $\ \|A^{j}{\boldsymbol{x}}\bigr{\|}\ \dot{\leq}\ \rho^{\,j}(A|_{L}),\
j\in{\mathbb{N}}$. Moreover, there exists ${\boldsymbol{x}}_{0}\in\Gamma$ and
$C>0$ such that $\
\|A^{j}{\boldsymbol{x}}_{0}\bigr{\|}\,\geq\,C\,\rho^{\,j}(A|_{L}),\
j\in{\mathbb{N}}$.

Proof. It suffices to consider the case $L=K$, otherwise we restrict $A$ to
the corresponding face plane. Denote $\rho=\rho(A)$ and take the invariant
face $\bar{L}$ from Proposition 3.1. Since $\Gamma$ does not lie on a proper
invariant face of $K$, it follows that there exists
${\boldsymbol{x}}_{0}\,\in\,\Gamma\setminus\bar{L}$, $C>0$, for which
$\,\|A^{j}{\boldsymbol{x}}_{0}\bigr{\|}\,\geq\,\quad C\,\rho^{\,j}(A),\
j\in{\mathbb{N}}$. On the other hand,
$\,\|A^{j}{\boldsymbol{x}}\bigr{\|}\,\dot{\leq}\,\rho^{\,j}$ for all
${\boldsymbol{x}}\in L$, which completes the proof.

$\Box$

##  4 Cones of nonnegative polynomials

Consider a finite subset $\Omega\subset{\mathbb{Z}}^{n}$ symmetric about the
origin and containing zero. Denote by ${\mathcal{P}}_{\Omega}$ the space of
real-valued trigonometric polynomials on ${\mathbb{R}}^{n}$ with the spectrum
on $\Omega$:

${\mathcal{P}}_{\Omega}\quad=\quad\left\\{\
{\boldsymbol{p}}({\boldsymbol{\xi}})\ =\
\sum_{{\boldsymbol{k}}\in\Omega}\,p_{{\boldsymbol{k}}}\,e^{2\pi
i\,({\boldsymbol{k}},{\boldsymbol{\xi}})}\quad:\quad
p_{-{\boldsymbol{k}}}\,=\,p_{{\boldsymbol{k}}}\,\in{\mathbb{R}}\,,\
{\boldsymbol{k}}\in\Omega\ \right\\}.$

We usually assume that $\Omega$ is fixed and use the simple notation
${\mathcal{P}}={\mathcal{P}}_{\Omega}$. We have ${\rm dim}\,{\mathcal{P}}\ =\
\bigl{(}1+|\Omega|\bigr{)}/2$. Consider the set ${\mathcal{P}}_{+}$ of
nonnegative polynomials from ${\mathcal{P}}$, i.e.,
${\boldsymbol{p}}({\boldsymbol{\xi}})\geq 0,\
{\boldsymbol{\xi}}\in{\mathbb{R}}^{n}$. For a given closed set
$S\subset{\mathbb{R}}^{n}$, we denote

${\mathcal{P}}(S)\quad=\quad\Bigl{\\{}\,{\boldsymbol{p}}\in{\mathcal{P}}_{\Omega}\
:\ {\boldsymbol{p}}({\boldsymbol{\xi}})=0,\ {\boldsymbol{\xi}}\in S\
\Bigr{\\}}\,,\qquad{\mathcal{P}}_{+}(S)\,=\,{\mathcal{P}}(S)\cap{\mathcal{P}}_{+}\,.$

The same notation is used for compact sets on the period:
$\,S\subset{\mathbb{T}}_{n}$. Clearly, the sets ${\mathcal{P}}(S)$ are linear
subspaces of ${\mathcal{P}}$ and ${\mathcal{P}}_{+}(S)$ are cones in those
subspaces.

###### Proposition 4.1

The set ${\mathcal{P}}_{+}$ is a closed solid pointed cone in ${\mathcal{P}}$.
Every face $L$ of ${\mathcal{P}}_{+}$ has the form $L={\mathcal{P}}_{+}(S)$
for a suitable compact subset $S\subset{\mathbb{T}}_{n}$.

Proof. The closeness and pointedness of ${\mathcal{P}}_{+}$ is obvious. To
prove that this cone is solid, one can note that for every
${\boldsymbol{q}}\in{\mathcal{P}}$, such that
$\|{\boldsymbol{q}}\|_{C({\mathbb{T}}_{n})}<1$, we have
$1+{\boldsymbol{q}}\in{\mathcal{P}}_{+}$, hence ${\mathcal{P}}_{+}$ contains
an open set. Now observe, that ${\mathcal{P}}_{+}$ is an intersection of the
half-spaces $\\{{\boldsymbol{p}}\in{\mathcal{P}}\ |\
{\boldsymbol{p}}({\boldsymbol{\xi}})\geq 0\\}$ over all
${\boldsymbol{\xi}}\in{\mathbb{T}}_{n}$. Therefore, every plane of support of
the cone ${\mathcal{P}}_{+}$ has the form
${\mathcal{P}}(\\{{\boldsymbol{\xi}}\\})\,=\,\bigl{\\{}\,{\boldsymbol{p}}\in{\mathcal{P}}\
:\ {\boldsymbol{p}}({\boldsymbol{\xi}})=0\bigr{\\}}$ and hence, every face
plane of ${\mathcal{P}}_{+}$ is an intersection of some of those planes. Thus,
every face of the cone ${\mathcal{P}}_{+}$ has the form ${\mathcal{P}}_{+}(S)$
for certain $S\subset{\mathbb{T}}_{n}$.

$\Box$

The proof of the following fact is the same as for Proposition 4.1:

###### Proposition 4.2

For every compact set $S\subset{\mathbb{T}}_{n}$, all faces of the cone
${\mathcal{P}}_{+}(S)$ are the sets ${\mathcal{P}}_{+}(S^{\prime}),\
S^{\prime}\subset S$.

For a given $\ell\geq 0$, we denote by ${\mathcal{P}}^{(\ell)}$ the set of
polynomials ${\boldsymbol{p}}\in{\mathcal{P}}$ that have zero of order at
least $2(\ell+1)$ at the origin, i.e., $\
{\boldsymbol{p}}({\boldsymbol{0}})={\boldsymbol{p}}^{\prime}({\boldsymbol{0}})=\cdots={\boldsymbol{p}}^{(2\ell+1)}({\boldsymbol{0}})=0$.
The equality ${\boldsymbol{p}}^{(k)}({\boldsymbol{x}})=0$ means that all
partial derivatives of order $k$ at the point ${\boldsymbol{x}}$ are equal to
zero.

We consider zeros of even order only, because nonnegative polynomials can have
only such zeros at the origin. The cone of nonnegative polynomials
${\mathcal{P}}^{(\ell)}_{+}\,=\,{\mathcal{P}}_{+}\cap{\mathcal{P}}^{(\ell)}$
is full-dimensional in ${\mathcal{P}}^{(\ell)}$. Similarly we define
${\mathcal{P}}^{(\ell)}_{+}(S)$ and prove the analogues of Propositions 4.1
and 4.2 for the space ${\mathcal{P}}^{(\ell)}$. We will refer to those
propositions mentioning that we use them for ${\mathcal{P}}^{(\ell)}$ instead
of ${\mathcal{P}}$. Sometimes we use the short notation $P_{\ell}=P^{(\ell)}$.

##  5 Regularity of refinable functions with an arbitrary dilation matrix

5.1 Preliminary facts and notation

We deal with the refinement equation (1) with complex coefficients
$\\{c_{{\boldsymbol{k}}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$ supported
on a finite set $Q\subset{\mathbb{Z}}^{n}$ and such that
$\sum_{{\boldsymbol{k}}\in Q}c_{{\boldsymbol{k}}}\,=\,m$. For the sake of
simplicity, we restrict ourselves to the case or real coefficients, although
all our results are directly extended to the complex case. The unique
compactly supported solution $\varphi\in{\mathcal{S}}_{0}^{\prime}$ (refinable
function) is normalized by the condition
$\int_{{\mathbb{R}}^{n}}\varphi\,d{\boldsymbol{x}}=1$. The equation satisfies
the sum rules of order $\ell\geq 0$ if the $m-1$ points
${\boldsymbol{\xi}}=M^{-T}{\boldsymbol{d}}_{k}^{*},\
{\boldsymbol{d}}_{k}^{*}\in D^{*}\setminus\\{{\boldsymbol{0}}\\}$ are roots of
order at least $\ell+1$ of the mask ${\boldsymbol{c}}({\boldsymbol{\xi}})$.
Recall that $D$ and $D^{*}$ are the sets of digits for the matrices $M,M^{T}$
respectively.

We make the standard assumption that the refinable function $\varphi$ is
stable i.e., its integer shifts
$\\{\varphi(\mathord{\,\cdot\,}-{\boldsymbol{k}})\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$
are linearly independent. Equivalently, those shifts form a Riesz basis of
their linear span. Without loss of generality we set
${\boldsymbol{d}}_{0}={\boldsymbol{d}}_{0}^{*}={\boldsymbol{0}}$. It is known
that if the solution $\varphi$ is stable and belongs to
$W^{\ell}_{2}({\mathbb{R}}^{n})$, then the refinement equation satisfies the
sum rules of order $\ell$. In particular, if $\varphi\in
L_{2}({\mathbb{R}}^{n})$, then the sum rules of order zero are fulfilled,
i.e.,
${\boldsymbol{c}}\,\bigl{(}M^{-T}{\boldsymbol{d}}_{k}^{*}\bigr{)}\,=\,0,\
k=1,\ldots,m-1$. In terms of the coefficients, this condition reads:

$\sum_{{\boldsymbol{j}}\in{\mathbb{Z}}^{n}}\,c_{M{\boldsymbol{j}}-{\boldsymbol{d}}_{k}}\
=\ 1\,,\qquad k=0,\ldots,m-1\,.$ (5)

It will always be assumed that the sum rules of order zero are satisfied. Now
define several special subsets of ${\mathbb{R}}^{n}$ and of
${\mathbb{Z}}^{n}$. Let

$Y\quad=\quad\left\\{\ \sum_{j=1}^{\infty}M^{-j}\,{\boldsymbol{s}}_{j}\
:\quad{\boldsymbol{s}}_{j}\in Q\,,\ j\in{\mathbb{N}}\ \right\\}\,.$

This is a compact self-similar set generated by the affine operators
$A_{{\boldsymbol{s}}}{\boldsymbol{x}}\,=\,M^{-1}({\boldsymbol{x}}+{\boldsymbol{s}}),\,{\boldsymbol{s}}\in
Q$. This means that $Y$ is a unique fixed point of the set-valued map
$AX\,=\,\cup_{{\boldsymbol{s}}\in Q}A_{{\boldsymbol{s}}}X$, i.e., an affine
fractal. The transition operator ${\mathcal{F}}$ respects the space of
functions supported on the set $Y$, therefore, ${\rm supp}\,\varphi\,\subset
Y$, see [4, 6] for details. Let
$\Omega\,=\,\bigl{(}Y-Y\bigr{)}\cap{\mathbb{Z}}^{n}$, where
$A-B\,=\,\\{{\boldsymbol{a}}-{\boldsymbol{b}}\,:\ {\boldsymbol{a}}\in
A,{\boldsymbol{b}}\in B\\}$ is the standard Minkowski subtractions of two
sets. Thus, $\Omega$ is a symmetric finite (since $Y$ is compact) subset of
${\mathbb{Z}}^{n}$ containing zero. This set is uniquely defined by the
support of the mask $Q$ and can be constructed by an algorithm presented in
Appendix. It follows from the definition that
$\Omega\,=\,\\{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}\,:\
Y\cap(Y+{\boldsymbol{k}})\,\neq\,\emptyset\\}$. Therefore, $\Omega$ contains
all integer points from the support of the function $\varphi*(\varphi\\_)$
(the symmetrized autoconvolution of $\varphi$), where
$\varphi\\_(\mathord{\,\cdot\,})=\varphi(-\mathord{\,\cdot\,})$.

We consider the linear operator ${\mathcal{T}}$ defined in
$L_{2}({\mathbb{T}}_{n})$ by formula (3). This operator maps the space of
polynomials ${\mathcal{P}}_{\Omega}$ to itself [15, 18, 26] and in the sequel
we restrict ${\mathcal{T}}$ to this subspace. If the mask ${\boldsymbol{c}}$
satisfies the sum rules of order $\ell$, then ${\mathcal{T}}$ respects also
all the subspaces ${\mathcal{P}}^{(k)},\,k\,=\,0,1,\ldots,\ell$, see [25, 27].
Clearly, it respects the positive cones ${\mathcal{P}}^{(k)}_{+}\,$ as well.
For an arbitrary compactly supported distribution
$f\in{\mathcal{S}}^{\prime}_{0}$, the periodization of the square of its
Fourier transform $\Phi_{f}$ defined in (2) is a non-negative trigonometric
polynomial with coefficients
$p_{{\boldsymbol{k}}}\,=\,[\,f*f\\_\,](-{\boldsymbol{k}})$. So, if ${\rm
supp}\,f\,\subset Y$, then $\Phi_{f}\,\in\,{\mathcal{P}}_{+}$. For every
$f\in{\mathcal{S}}_{0}^{\prime}$, we have
${\mathcal{T}}(\Phi_{f})\,=\,\Phi_{{\mathcal{F}}f}$.

For each ${\boldsymbol{p}}\in{\mathcal{P}}_{+}$, we are interested in the
asymptotic behaviour of the sequence
$\\{{\mathcal{T}}^{k}{\boldsymbol{p}}\\}_{k\in{\mathbb{N}}}$ as $k\to\infty$.
In view of Theorem 3.3, it is defined by the spectral radius of the operator
${\mathcal{T}}$ restricted to the minimal invariant face of the cone
${\mathcal{P}}_{+}$ that contains ${\boldsymbol{p}}$. That face, in turn, is
determined by the set of zeros $S$ of the polynomial ${\boldsymbol{p}}$
(Proposition 4.1). This set $S$ must possess a special structure because the
space of polynomials that vanish on $S$ has to be invariant with respect to
the operator ${\mathcal{T}}$. We need to reveal this structure and find this
set.

For arbitrary $s=1,\ldots,q$, we consider the set $\,\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}\,$ that consists of
points ${\boldsymbol{\xi}}\in{\mathbb{R}}^{n}$ such that
${\boldsymbol{\xi}}+{\boldsymbol{k}}\ \in\ \mbox{\em
nill}\,(\widehat{\varphi})\,\cup\,J_{s}^{\perp}$ for all
${\boldsymbol{k}}\in{\mathbb{Z}}^{d}$. Let us remember that $\mbox{\em
nill}\,(f)$ denotes the set of zeros of an analytic function $f$ and
$J^{\perp}$ is an orthogonal complement to the subspace $J$ in
${\mathbb{R}}^{n}$. Thus, the periodized point ${\boldsymbol{\xi}}$ is
contained in the union of the sets $\mbox{\em nill}\,(\widehat{\varphi})$ and
$J_{s}^{\perp}$. The stability of the function $\varphi$ implies that its
Fourier transform $\widehat{\varphi}$ does not have periodic zeros. This
yields that $\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\varphi})\bigr{\\}}\,=\,\emptyset$. So, only the hyperplane
$J_{s}^{\perp}$ makes this set nonempty. Indeed, the set $\mbox{\em
nill}\,(\widehat{\varphi})$ contains all integer points except for zero while
$J_{s}^{\perp}$ contains zero, consequently,
${\boldsymbol{0}}\in\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$.

Denote by $P_{\ell,s}$ the space of polynomials from ${\mathcal{P}}_{\Omega}$
that have zeros of order at least $2(\ell+1)$ at all points of the set
$\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}\,$ (see
subsection 1.4 for the notation). The cone of nonnegative polynomials in
$P_{\ell,s}$ is referred to as $K_{\ell,s}$. This is a face of the cone
${\mathcal{P}}_{+}$ with the corresponding face plane $P_{\ell,s}$.

###### Lemma 5.1

If the refinement equation satisfies sum rules of order $\ell$, then for every
$s=1,\ldots,q$, the space $P_{\ell,s}$ is an invariant subspace of the
operator ${\mathcal{T}}$ and $K_{\ell,s}$ is its invariant cone in this
subspace.

Proof is in Appendix.

###### Proposition 5.2

Let a refinement equation satisfy the sum rules of order $\ell$. Then, for
each $s=1,\ldots,q$, there is an alternative: either
$P_{\ell,s}={\mathcal{P}}^{(\ell)}$ or there exists a non-integer vector
${\boldsymbol{z}}\in{\mathbb{R}}^{n}$ orthogonal to $J_{s}$ such that for
every ${\boldsymbol{k}}\in{\mathbb{Z}}^{n}$, the vector
${\boldsymbol{z}}+{\boldsymbol{k}}$ is orthogonal to $J_{s}$ or is a zero of
the function $\widehat{\varphi}$ of order at least $\ell+1$.

Proof. As we already noted, the set $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}\,$ always
contains $\\{{\boldsymbol{0}}\\}$. On the other hand, all polynomials from
${\mathcal{P}}^{(\ell)}$ have a zero of order $\,\geq\,2(\ell+1)$ at the point
${\boldsymbol{0}}$. Hence, $P_{\ell,s}\,\neq\,{\mathcal{P}}^{(\ell)}$ if and
only if the set $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}\,$ contains a
non-integer point ${\boldsymbol{z}}$. Since the function $\varphi$ is stable,
it follows that $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\bigr{\\}}\,=\,\emptyset$. Therefore,
${\boldsymbol{z}}\in J_{s}^{\perp}$, which concludes the proof.

$\Box$

5.2. The $L_{2}$-regularity

Now we prove the formula for the $L_{2}$-regularity and in the next section
attack the higher order regularity. The main idea is to work separately with
regularities along spectral subspaces $J_{1},\ldots,J_{q}$ of $M$. Then the
regularity of $\varphi$ is equal to the minimal regularity over these
subspaces.

We denote by $\alpha_{0}(f)$ the Hölder exponent (or the Hölder exponent of
zero order) of a function $f\in L_{2}({\mathbb{R}}^{n})$. The index zero is
added to distinguish from the Hölder exponent of higher order.

For a given nontrivial subspace $J\subset{\mathbb{R}}^{n}$ (the case
$J={\mathbb{R}}^{n}$ is allowed), we define the Hölder exponent in $L_{2}$ of
the function $f\in L_{2}({\mathbb{R}}^{n})$ along $J$:

$\alpha_{\,0,J}\quad=\quad\sup\ \Bigl{\\{}\alpha\geq 0\ :\
\|f(\mathord{\,\cdot\,}+{\boldsymbol{h}})-f(\mathord{\,\cdot\,})\|\ \leq\
C\,\|{\boldsymbol{h}}\|^{\,\alpha}\,,\ {\boldsymbol{h}}\in J\Bigr{\\}}\,.$

For the subspaces $J_{s}$, we use the short notation
$\alpha_{\,0,s}\,=\,\alpha_{\,0,J_{s}}\,,\ s=1,\ldots,q$.

The spectral radius of the operator ${\mathcal{T}}$ restricted to its
invariant subspace $P_{0,s}$ is denoted by $\rho_{\,0,s}$.

###### Theorem 5.3

For an arbitrary refinable function $\varphi$ with the dilation matrix $M$, we
have

$\alpha_{\,0,s}(\varphi)\quad=\quad\frac{1}{2}\,\log_{1/r_{s}}\,\rho_{\,0,s}\,,\qquad
s=1,\ldots,q,$ (6)

where $J_{s}$ are the spectral subspaces of $M$. For the Hölder exponent in
$L_{2}({\mathbb{R}}^{n})$, we have

$\alpha_{0}(\varphi)\quad=\quad\frac{1}{2}\,\min_{s=1,\ldots,q}\log_{\,1/r_{s}}\,\rho_{\,0,s}$
(7)

Note that $P_{0,1}$ can be a proper subspace of ${\mathcal{P}}^{(0)}$ only if
the mask possesses a special set of zeros. That is why, for most of refinement
equations, we have $P_{0,1}={\mathcal{P}}^{(0)}$, and hence, the spectral
radius $\rho_{\,0,1}$ coincides with
$\rho_{0}\,=\,\rho\,\bigl{(}{\mathcal{T}}|_{{\mathcal{P}}^{(0)}}\bigr{)}$.
Since $r=r_{1}$ is the maximal spectral radius among all
$r_{s},\,s=1,\ldots,q$, we see that the minimum in the formula (7) is attained
for $s=1$. Therefore,
$\alpha_{0}(\varphi)\,=\,\frac{1}{2}\,\log_{\,1/r_{1}}\,\rho_{0}$. Applying
Proposition 5.2 for the subspace $J_{1}$, we obtain

###### Theorem 5.4

The equality $\alpha_{0}(\varphi)\,=\,\frac{1}{2}\,\log_{1/r}\,\rho_{0}$
holds, unless there exists a non-integer point ${\boldsymbol{z}}\in
J_{1}^{\perp}$ such that, for every integer shift
${\boldsymbol{z}}+{\boldsymbol{k}},\,{\boldsymbol{k}}\in{\mathbb{Z}}^{n}$, we
have either ${\boldsymbol{z}}+{\boldsymbol{k}}\,\in\,J_{1}^{\perp}$ or
$\widehat{\varphi}({\boldsymbol{z}}+{\boldsymbol{k}})=0$.

There are, however, many examples when $P_{0,1}$ is a proper subspace of
${\mathcal{P}}^{(0)}$ and, respectively, $\rho_{\,0,1}<\rho_{0}$. In this
case, the Hölder exponent can be larger than
$\frac{1}{2}\,\log_{\,1/r_{1}}\,\rho_{0}$ and is equal to the minimal value
$\,\frac{1}{2}\,\log_{\,1/r_{s}}\,\rho_{\,0,s}$ among $s\,=\,2,\ldots,q$.

The proof of Theorem 5.4 requires several further notation and auxiliary
results. In the sequel of this section we use the simplified notation
$P=P_{0}$ (the subspace of polynomials from ${\mathcal{P}}_{\Omega}$ such that
${\boldsymbol{p}}(0)={\boldsymbol{p}}^{\prime}(0)=0$), $P_{s}=P_{0,s}$ (the
space of polynomials from $P$ that vanish on the set $\pi\\{\mbox{\em
nill}\,(\widehat{\varphi}),J_{s}^{\perp}\\}$). We also denote $K=K_{0}$ and
$K_{s}=K_{0,s}$, those are the cones of nonnegative polynomials in $P$ and in
$P_{s}$ respectively.

An $\varepsilon$-extension $Y_{\varepsilon}$ of $Y$ is a compact set
containing a neighbourhood of $Y$ such that $Y_{\varepsilon}\subset
Y+B(0,\varepsilon)$ and $AY_{\varepsilon}\subset Y_{\varepsilon}$, where
$AY_{\varepsilon}\,=\,\cup_{{\boldsymbol{s}}\in
Q}A_{{\boldsymbol{s}}}Y_{\varepsilon}$. Let us remember that $AY=Y$. For every
$\varepsilon$-extension $Y_{\varepsilon}$, the transition operator
${\mathcal{F}}$ respects the space of functions supported on the set
$Y_{\varepsilon}$.

###### Lemma 5.5

For every $\varepsilon>0$, there exists an $\varepsilon$-extension of $Y$.

Proof. Take a number $t>0$ such that the diameter $tY$ is less than
$\varepsilon$ and define $Y_{\varepsilon}=Y+tY$ (the Minkowski sum). This set
contains a neighbourhood of $Y$ because $tY$ has a nonempty interior and,
clearly, $AY_{\varepsilon}\subset Y_{\varepsilon}$.

$\Box$

###### Remark 5.6

If $Y$ is not convex, then $Y+tY$ is not necessarily equal to $(1+t)Y$.

Let us remember that for every $f\in{\mathcal{S}}_{0}^{\prime}$, the function
$\Phi_{f}$ is a trigonometric polynomial and if ${\rm supp}\,f\,\subset\,Y$,
then $\Phi_{f}\in{\mathcal{P}}_{\Omega}$, where
$\Omega=(Y-Y)\cap{\mathbb{Z}}^{n}$. Since $Y$ is compact, it follows that a
sufficiently small enlargement of the set $Y$ does not change the set
$\Omega$. Thus, the set
$(Y_{\varepsilon}-Y_{\varepsilon})\cap{\mathbb{Z}}^{n}$ coincides with
$\Omega$, whenever $\varepsilon$ is small enough. Fix such $\varepsilon$ and
an $\varepsilon$-extension $Y_{\varepsilon}$. For each $s=1,\ldots,q$,
consider the set of differences of the function $\varphi$ under small shifts
along $J_{s}$:

$U_{s}\ =\
\Bigl{\\{}f(\cdot)\,=\,\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})\,-\,\varphi(\mathord{\,\cdot\,})\
:\quad{\boldsymbol{h}}\in J_{s}\,,\ {\rm
supp}\,f\,\subset\,Y_{\varepsilon}\,\Bigr{\\}}\,.$ (8)

Those sets are invariant with respect to the transition operator
${\mathcal{F}}$. Observe that $U_{s}$ are not linear subspaces. Their linear
spans may have nontrivial intersections, may even all coincide, while, recall,
the spaces $J_{s},\,s=1,\ldots,q$, form a decomposition of ${\mathbb{R}}^{n}$
into a direct sum. Every $f\in U_{s}$ is defined by the translation vector
${\boldsymbol{h}}\in J_{s}$, that is why we sometimes denote
$f=f_{{\boldsymbol{h}}}$.

Now we are ready to prove Theorem 5.3.

Proof of Theorem 5.3. Consider the space $J_{s}$ for some $s=1,\ldots,q$. For
an arbitrary function $f\in U_{s}$, the Plancherel theorem and the equality
$\Phi_{\,{\mathcal{F}}f}\,=\,{\mathcal{T}}\,\Phi_{f},\,f\in{\mathcal{S}}_{0}^{\prime}$,
imply

$\bigl{\|}{\mathcal{F}}^{j}\,f\bigr{\|}_{2}^{2}\quad=\quad\bigl{\|}\widehat{{\mathcal{F}}^{j}\,f}\bigr{\|}_{2}^{2}\quad=\quad\bigl{\|}{\mathcal{T}}^{j}\Phi_{f}\,\bigr{\|}_{1}\,.$
(9)

Moreover, the function $\Phi_{f}$ is a polynomial from ${\mathcal{P}}_{+}$.
Let $V_{s}$ be the minimal face of the cone ${\mathcal{P}}_{+}$ containing the
set $\Gamma=\\{\Phi_{f}\,:\,\ f\in U_{s}\\}$. Since ${\mathcal{F}}U_{s}\subset
U_{s}$, it follows that ${\mathcal{T}}V_{s}\subset V_{s}$. Let us show that
$V_{s}=K_{s}$. For each $f\in U_{s}$, the Fourier transform
$\widehat{f}({\boldsymbol{\xi}})\,=\,\bigl{(}e^{2\pi
i({\boldsymbol{h}},{\boldsymbol{\xi}})}\,-\,1\bigr{)}\widehat{\varphi}({\boldsymbol{\xi}})$
vanishes on the family of parallel hyperplanes
$\\{{\boldsymbol{\xi}}\in{\mathbb{R}}^{n}\,|\,({\boldsymbol{h}},{\boldsymbol{\xi}})\in{\mathbb{Z}}\\}$.
This property holds for all sufficiently small ${\boldsymbol{h}}\in J_{s}$
precisely when $f$ vanishes on the set of all sufficiently small vectors
${\boldsymbol{\xi}}\in{\mathbb{R}}^{n}$ orthogonal to the subspace $J_{s}$.
This means that it vanishes on the orthogonal complement of $J_{s}$ in
${\mathbb{R}}^{n}$. Each $\Phi_{f}\in\Gamma$ generating $V_{s}$ vanishes on
the union of the sets $\mbox{\em nill}\,(\widehat{\varphi})$ and
${\boldsymbol{h}}^{\perp}$. The intersection of all those sets over all
feasible $\,{\boldsymbol{h}}\in J_{s}$ is $\mbox{\em
nill}\,\,(\widehat{\varphi})\,\cup\,J^{\perp}_{s}$. By Proposition 4.1, every
face of ${\mathcal{P}}_{+}$ consists of nonnegative polynomials from
${\mathcal{P}}_{\Omega}$ that vanish on a certain closed set
$S\subset{\mathbb{R}}^{n}$. Therefore, the face $V_{s}$ is the set of
polynomials from ${\mathcal{P}}_{+}$ that vanish on the set $\pi\,\\{\mbox{\em
nill}\,(\widehat{\varphi}),J_{s}^{\perp}\\}$, i.e., $V_{s}=K_{s}$.

Since all eigenvalues of the matrix $M$ restricted to the subspace $J_{s}$ are
equal to $r_{s}$, it follows that there exists a constant $c_{1}>0$ such that

$c_{1\,}r_{s}^{j}\
\|{\boldsymbol{h}}\|\quad\leq\quad\|M^{j}{\boldsymbol{h}}\|\quad\dot{\leq}\quad
r_{s}^{j}\,\|{\boldsymbol{h}}\|\,,\qquad{\boldsymbol{h}}\in J_{s},\
j\,\in\,{\mathbb{N}}\,.$

Hence, for every $\delta>0$, there exists a constant $c_{2}$ such that
$\|M^{j}{\boldsymbol{h}}\|\ \leq\
c_{2}\,(r_{s}+\delta)^{j}\,\|{\boldsymbol{h}}\|,\,j\in{\mathbb{N}}$. Further,
choose some $\tau<\varepsilon$ such that the $\tau$-neighbourhood of $Y$ is
contained in $Y_{\varepsilon}$. Take arbitrary ${\boldsymbol{h}}\in
J_{s},\,\|{\boldsymbol{h}}\|<\tau$, and the smallest integer $j$ such that
$\|M^{j+1}{\boldsymbol{h}}\|\,\geq\,\tau$. Since
$\|M^{j}{\boldsymbol{h}}\|\,<\,\tau$, it follows that the function
$f=\varphi(\mathord{\,\cdot\,}+M^{j}{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})$
belongs to $U_{s}$. Applying (9), we obtain

$\Bigl{\|}\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})\,-\,\varphi(\mathord{\,\cdot\,})\Bigr{\|}^{2}\
=\
\Bigl{\|}{\mathcal{F}}^{j}\,\bigl{(}\varphi(\mathord{\,\cdot\,}+M^{j}{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})\bigr{)}\Bigr{\|}^{2}\
=\ \Bigl{\|}{\mathcal{F}}^{j}\,f\Bigr{\|}^{2}\ =\
\Bigl{\|}{\mathcal{T}}^{j}\Phi_{f}\,\Bigr{\|}_{1}\,.$

On the other hand, $K_{s}$ is the minimal invariant face of
${\mathcal{P}}_{+}$ that contains $\Gamma$. By Theorem 3.3, for every
$\Phi_{f}\in\Gamma$, we have
$\bigl{\|}{\mathcal{T}}^{j}\Phi_{f}\,\bigr{\|}_{1}\,\dot{\leq}\,(\rho_{\,0,s})^{\,j}\|\Phi_{f}\|_{1}$.
Furthermore,
$\|\Phi_{f}\|_{1}\,=\,\|\widehat{f}\|^{2}\,=\,\|f\|^{2}\,\leq\,4\|\varphi\|^{2}$,
because
$\|f\|\,=\,\|\varphi(\mathord{\,\cdot\,}+M^{j}{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})\|\,\leq\,2\,\|\varphi\|.$
Thus,

$\bigl{\|}\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})\,-\,\varphi(\mathord{\,\cdot\,})\bigr{\|}^{2}\quad\dot{\leq}\quad(\rho_{\,0,s})^{\,j}\,.$
(10)

On the other hand, $\ \tau\,\leq\,\|M^{j+1}{\boldsymbol{h}}\|\ \leq\
c_{2}(r_{s}+\delta)^{\,j+1}\|{\boldsymbol{h}}\|$. Therefore,
$\|{\boldsymbol{h}}\|\,\geq\,\frac{\tau}{c_{2}}\,(r_{s}+\delta)^{j+1}$.
Combining this with (10) and taking the limit as $j\to\infty$, we obtain
$\,\alpha_{\,0,s}\,\geq\,\frac{1}{2}\,\log_{1/(r_{s}+\delta)}\,\rho_{\,0,s}$.
This is true for every $\delta>0$, consequently,
$\,\alpha_{\,0,s}\,\geq\,\frac{1}{2}\,\log_{1/r_{s}}\,\rho_{\,0,s}$.

To establish the inverse inequality we invoke Theorem 3.3: there exists
$\Phi_{f}\in\Gamma$ such that $\|{\mathcal{T}}^{j}\Phi_{f}\|\ \geq\
C\,(\rho_{\,0,s})^{j}$. Applying (9), we get
$\,\bigl{\|}{\mathcal{F}}^{j}\,f\bigr{\|}^{2}\,=\,\|{\mathcal{T}}^{j}\Phi_{f}\|\,\geq\,C\,(\rho_{\,0,s})^{j}$.
Since
$f(\mathord{\,\cdot\,})\,=\,\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})$
for some ${\boldsymbol{h}}\in J_{s}$, we see that

$\bigl{\|}\varphi(\mathord{\,\cdot\,}+M^{-j}{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,})\bigr{\|}\
=\
\bigl{\|}{\mathcal{F}}^{j}(\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\mathord{\,\cdot\,}))\bigr{\|}\
=\ \bigl{\|}{\mathcal{F}}^{j}f\bigr{\|}\ \geq\ C\,(\rho_{\,0,s})^{j/2}\,.$

On the other hand,
$\,\|M^{-j}{\boldsymbol{h}}\|\,\leq\,c_{1}\,(r_{s}-\delta)^{-j}\|{\boldsymbol{h}}\|$,
which implies that
$\alpha_{\,0,s}\,\leq\,\frac{1}{2}\,\log_{1/(r_{s}-2\delta)}\,\,\rho_{\,0,s}$.
Taking the limit as $\delta\to 0$, we conclude the proof.

$\Box$

5.3 The higher order regularity

For an integer $k\geq 0$ and for a given nontrivial subspace
$J\subset{\mathbb{R}}^{n}$, we denote by $W^{k}_{2}({\mathbb{R}}^{n},J)$ the
space of functions $k$ times differentiable along $J$. This means that for
every multivector
$\bar{\boldsymbol{b}}\,=\,({\boldsymbol{b}}_{1},\ldots,{\boldsymbol{b}}_{k})\in
J^{\,k}$, where $J^{\,k}\,=\,J\times\cdots\times J$ ($k$ multipliers),
${\boldsymbol{b}}_{i}\neq 0,\,i=1,\ldots,k$, the mixed partial derivative
$f_{\bar{\boldsymbol{b}}}\,=\,\frac{\partial}{\partial{\boldsymbol{b}}_{1}\cdots\partial{\boldsymbol{b}}_{k}}\,f$
along the vectors ${\boldsymbol{b}}_{1},\ldots,{\boldsymbol{b}}_{k}$ belongs
to $L_{2}({\mathbb{R}}^{n})$. For every $f\in W^{k}_{2}({\mathbb{R}}^{n},J)$,
the Hölder exponent of order $k$ along $J$ is defined as follows:

$\alpha_{k,J}\quad=\quad k\ +\ \sup\ \Bigl{\\{}\alpha\geq 0\ :\
\|f_{\bar{\boldsymbol{b}}}(\mathord{\,\cdot\,}+{\boldsymbol{h}})-f_{\bar{\boldsymbol{b}}}(\mathord{\,\cdot\,})\|\
\leq\ C\,\|{\boldsymbol{h}}\|^{\,\alpha}\,,\ \bar{\boldsymbol{b}}\in
J^{\,k}\Bigr{\\}}\,.$

For the subspaces $\\{J_{s}\\}_{s=1}^{q}$ (the spectral subspaces of the
dilation matrix $M$), we use the short notation $\
\alpha_{k,J_{s}}\,=\,\alpha_{\,k,s}\,,\ s=1,\ldots,q$.

If $J={\mathbb{R}}^{n}$, then we obtain the standard definition of the Hölder
exponent of the $k$th order $\alpha_{k}(f)$ as the number $k$ plus the minimal
Hölder exponent (of order zero) of all mixed derivatives of order $k$. The
(generalized) Hölder exponent of a function $f\in L_{2}({\mathbb{R}}^{n})$ is
$\alpha(f)\,=\,\alpha_{\,k}(f)$, where $k$ is the largest integer, for which
$f\in W_{2}^{k}({\mathbb{R}}^{n})$.

From the representation by the direct sum
${\mathbb{R}}^{n}\,=\,\oplus\sum_{s=1}^{q}J_{s}$, it follows that if a
function $f$ belongs to $W_{2}^{k}({\mathbb{R}}^{n},J_{s})$ for each $s$, then
it belongs to $W_{2}^{k}({\mathbb{R}}^{n})$ and the Hölder exponent
$\alpha_{k}(f)$ is equal to $\min_{s=1,\ldots,q}\alpha_{\,k,s}$. The
computation of each exponent of regularity $\alpha_{\,k,s}$ will be done
similarly to the previous section, in terms of spectral radii of the operator
${\mathcal{T}}$ on faces of the cone of nonnegative polynomials
${\mathcal{P}}_{+}$.

In the statement of the main result we invoke the notation of the space
$P_{k,s}$, the cone $K_{k,s}$, and the spectral radius as $\rho_{k,s}$ of the
operator ${\mathcal{T}}|_{P_{k,s}}$, see subsection 5.1. Recall that $P_{k,s}$
is respected by ${\mathcal{T}}$, provided the refinable function $\varphi$
satisfies the sum rules of order $k$ (Lemma 5.1). We also use the short
notation $P_{k}=P^{(k)}$.

###### Theorem 5.7

A refinable function $\varphi$ satisfying the sum rules of order $k$ belongs
to $W^{k}_{2}({\mathbb{R}}^{n},J_{s})$ precisely when
$\frac{1}{2}\,\log_{1/r_{s}}\rho_{k,s}\,>\,k$. In this case
$\alpha_{\,k,s}(\varphi)\,=\,\frac{1}{2}\,\log_{1/r_{s}}\rho_{k,s}$.

The function $\varphi$ belongs to $W_{2}^{k}({\mathbb{R}}^{n})$ if and only it
satisfies the sum rules of order $k$ and belongs to
$W^{k}_{2}({\mathbb{R}}^{n},J_{s})$ for all $s=1,\ldots,q$. The Hölder
exponent of $\varphi$ is equal to

$\alpha(\varphi)\ =\
\min_{s=1,\ldots,q}\,\frac{1}{2}\,\log_{1/r_{s}}\rho_{k,s}\,,$ (11)

where $k$ is the maximal number such that $\varphi\in
W^{k}_{2}({\mathbb{R}}^{n})$.

Applying Proposition 5.2 for the subspace $J_{1}$, we obtain

###### Theorem 5.8

The equality $\alpha(\varphi)\,=\,\frac{1}{2}\,\log_{1/r}\,\rho_{k}$, where
$r=\rho(M),\,\rho_{k}=\rho({\mathcal{T}}|_{P_{k}})$ and $k$ is the maximal
number such that $\frac{1}{2}\,\log_{1/r}\,\rho_{k}\,>\,k$, holds, unless
there exists a non-integer point ${\boldsymbol{z}}\in J_{1}^{\perp}$ such that
every integer shift ${\boldsymbol{z}}+{\boldsymbol{k}}\,,\
{\boldsymbol{k}}\in{\mathbb{Z}}^{n}$ is either on $J_{1}^{\perp}$ or is a zero
of the function $\widehat{\varphi}$ of order at least $k+1$.

The proof of Theorem 5.8 is realized in the same way as the proof of Theorem
5.3, but replacing the difference
$\varphi(\mathord{\,\cdot\,}+{\boldsymbol{h}})-\varphi(\cdot)$ by the higher
order differences $\Delta_{\,\bar{\boldsymbol{h}}}\varphi$. We begin with
necessary definitions.

For a given multivector
$\bar{\boldsymbol{h}}=({\boldsymbol{h}}_{0},\ldots,{\boldsymbol{h}}_{k})\in({\mathbb{R}}^{n}\setminus\\{0\\})^{k+1}$,
we denote by $\Delta_{\,\bar{\boldsymbol{h}}}f(\mathord{\,\cdot\,})$ the
$(k+1)$st order difference along the vectors ${\boldsymbol{h}}_{i}$. It is
defined inductively as

$\Delta_{\,({\boldsymbol{h}}_{0},\ldots,{\boldsymbol{h}}_{i})}f(\mathord{\,\cdot\,})\
=\
\Delta_{\,({\boldsymbol{h}}_{0},\ldots,{\boldsymbol{h}}_{i-1})}f(\mathord{\,\cdot\,}+{\boldsymbol{h}}_{i})\
-\
\Delta_{\,({\boldsymbol{h}}_{0},\ldots,{\boldsymbol{h}}_{i-1})}f(\mathord{\,\cdot\,})\
,\qquad i=1,\ldots,k\,.$

n what follows we always assume that all the vectors of shifts
${\boldsymbol{h}}_{i}$ are nonzero and denote
$\|\bar{\boldsymbol{h}}\|\,=\,\max_{i=0,\ldots,k}\|{\boldsymbol{h}}_{i}\|$. We
need the following well-known fact from the approximation theory [31, 35], see
also [48, Theorem 4.6.14]. It is usually formulated for $J={\mathbb{R}}^{n}$,
but the proofs are the same for proper subspaces $J\subset{\mathbb{R}}^{n}$.

Theorem A. For an arbitrary function $f\in W^{k}_{2}({\mathbb{R}}^{n},J)$, we
have

$\|\Delta_{\,\bar{\boldsymbol{h}}}f\|\ =\
o(\|\bar{\boldsymbol{h}}\|^{k}),\qquad\bar{\boldsymbol{h}}\in J^{k+1},\
\|\bar{\boldsymbol{h}}\|\,\to\,0\,.$

Conversely, if
$\|\Delta_{\,\tilde{\boldsymbol{h}}}f\|\,=\,O\,\bigl{(}\|\tilde{\boldsymbol{h}}\|^{k}\bigr{)},\,\tilde{\boldsymbol{h}}\in
J^{k},\,\|\tilde{\boldsymbol{h}}\|\,\to\,0$, then $f$ belongs to
$W^{k}_{2}({\mathbb{R}}^{n},J)$ after possible correction on a set of zero
measure. In this case

$\alpha_{\,k,J}(f)\quad=\quad\sup\ \Bigl{\\{}\,\alpha>0\ :\
\|\Delta_{\,\bar{\boldsymbol{h}}}f\|\,\leq\,C\,\|\bar{\boldsymbol{h}}\|^{\,\alpha},\qquad\bar{\boldsymbol{h}}\in
J^{k+1}\,\Bigr{\\}}\,.$ (12)

We use the $\varepsilon$-extensions $Y_{s}$ of the set $Y$ (Lemma 5.5) and the
following high-order analogues of the sets $U_{s}$ from (8):

$U_{k,s}\ =\
\Bigl{\\{}\,f=\Delta_{\,\bar{\boldsymbol{h}}}\varphi\,:\quad\bar{\boldsymbol{h}}\in
J_{s}^{\,k+1}\,,\ {\rm supp}\,f\,\subset\,Y_{\varepsilon}\,\Bigr{\\}}\,.$ (13)

Thus, $U_{k,s}$ is the set of $(k+1)$st order differences of the refinable
function $\varphi$ under small shifts along $J_{s}$. All those sets are
invariant with respect to the transition operator ${\mathcal{F}}$.

Proof of Theorem 5.7. For arbitrary $f\in U_{k,s}$, the function $\Phi_{f}$ is
a polynomial from ${\mathcal{P}}_{+}$. We denote by $V_{k,s}$ the minimal face
of the cone ${\mathcal{P}}_{+}$ containing the set
$\Gamma\,=\,\\{\Phi_{f}\,:\,\ f\in U_{k,s}\\}$. Since
${\mathcal{T}}\,\Gamma\subset\Gamma$, it follows that
${\mathcal{T}}V_{k,s}\subset V_{k,s}$. To show that $V_{k,s}=K_{k,s}$ we
observe that the Fourier transform of every function $f\in U_{k,s}$ has the
form
$\widehat{f}({\boldsymbol{\xi}})\,=\,\widehat{\varphi}({\boldsymbol{\xi}})\prod_{a=0}^{k}\,\bigl{(}e^{2\pi
i({\boldsymbol{h}}_{a},{\boldsymbol{\xi}})}\,-\,1\bigr{)}$ and hence, vanishes
on each family of parallel hyperplanes
$\\{{\boldsymbol{\xi}}\in{\mathbb{R}}^{n}\,|\,({\boldsymbol{h}}_{a},{\boldsymbol{\xi}})\in{\mathbb{Z}}\\},\
a=0,\ldots,k$. This property holds for all sufficiently small
${\boldsymbol{h}}_{a}\in J_{s}$ precisely when $f$ has zero of order at least
$k+1$ at each point of the subspace $J_{s}^{\perp}$. Therefore, each
polynomial $\Phi_{f}$ has zeros of order at least $2(k+1)$ on the union of the
sets $\mbox{\em nill}\,_{k}(\widehat{\varphi})$ and
${\boldsymbol{h}}_{a}^{\perp}$. The intersection of all those sets over all
feasible ${\boldsymbol{h}}_{a}\in J_{s},\,a=0,\ldots,k$, is $\mbox{\em
nill}\,_{k}(\widehat{\varphi})\cup J^{\perp}_{s}$. By Proposition 4.1, the
face $V_{k,s}$ coincides with the set of polynomials from ${\mathcal{P}}_{+}$
that have zeros of order at least $2(k+1)$ on the set $\pi\,\\{\mbox{\em
nill}\,_{k}(\widehat{\varphi}),J_{s}^{\perp}\\}$, which is $K_{k,s}$.

The remainder of the proof is the same as for Theorem 5.3. Choose some
$\tau<\varepsilon$ such that the $(k+1)\tau$-neighbourhood of $Y$ is contained
in $Y_{\varepsilon}$. Denote
$M^{j}\bar{\boldsymbol{h}}\,=\,\bigl{(}M^{j}{\boldsymbol{h}}_{0},\ldots,M^{j}{\boldsymbol{h}}_{k}\bigr{)}$.
For arbitrary $\bar{\boldsymbol{h}}\in
J_{s}^{k+1},\,\|\bar{\boldsymbol{h}}\|<\tau$, we find the biggest integer $j$
such that $\|M^{j+1}\bar{\boldsymbol{h}}\|\,\geq\,\,\tau$. The function
$f\,=\,\Delta_{\,M^{j}\bar{\boldsymbol{h}}}\,\varphi\,$ belongs to $U_{k,s}$,
and hence, applying (9), we obtain

$\Bigl{\|}\Delta_{\,\bar{\boldsymbol{h}}}\varphi\Bigr{\|}^{2}\ =\
\Bigl{\|}{\mathcal{F}}^{j}\,\Delta_{\,M^{j}\bar{\boldsymbol{h}}}\varphi\Bigr{\|}^{2}\
=\ \Bigl{\|}{\mathcal{F}}^{j}\,f\Bigr{\|}^{2}\ =\
\Bigl{\|}{\mathcal{T}}^{j}\Phi_{f}\,\Bigr{\|}_{1}\,.$

On the other hand, $\Phi_{f}$ belongs to the set $\Gamma$, for which $K_{k,s}$
is the minimal invariant face. By Theorem 3.3, we have
$\bigl{\|}{\mathcal{T}}^{j}\Phi_{f}\,\bigr{\|}_{1}\,\dot{\leq}\,(\rho_{\,0,s})^{\,j}\|\Phi_{f}\|_{1}$,
and consequently
$\bigl{\|}\Delta_{\,\bar{\boldsymbol{h}}}\varphi\bigr{\|}^{2}\,\dot{\leq}\,(\rho_{\,k,s})^{\,j}\,.$
Combining this with the inequality
$\|M^{j+1}\bar{\boldsymbol{h}}\|\,\geq\,\,\tau$, and taking $j\to\infty$, we
conclude that
$\,\alpha_{\,k,s}\,\geq\,\frac{1}{2}\,\log_{1/r_{s}}\,\rho_{\,k,s}$.

Further, by Theorem 3.3, there exists $\Phi_{f}\in\Gamma$ such that
$\|{\mathcal{T}}^{j}\Phi_{f}\|\ \geq\ C\,(\rho_{k,s})^{j}$. Applying (9), we
get
$\,\bigl{\|}{\mathcal{F}}^{j}\,f\bigr{\|}^{2}\,=\,\|{\mathcal{T}}^{j}\Phi_{f}\|\,\geq\,C\,(\rho_{k,s})^{j}$.
Since $f\,=\,\Delta_{\,\bar{\boldsymbol{h}}}\varphi$ for some
$\bar{\boldsymbol{h}}\in J_{s}^{\,k+1}$, we see that

$\bigl{\|}\Delta_{\,M^{-j}\bar{\boldsymbol{h}}}\varphi\bigr{\|}\ =\
\bigl{\|}{\mathcal{F}}^{j}\Delta_{\,\bar{\boldsymbol{h}}}\varphi\bigr{\|}\ =\
\bigl{\|}{\mathcal{F}}^{j}f\bigr{\|}\ \geq\ C\,(\rho_{\,0,s})^{j/2}\,.$

Combining this with the equality $\,\|M^{j}\bar{\boldsymbol{h}}\|\,<\,\,\tau$
and taking $j\to\infty$, we conclude that
$\alpha_{\,k,s}\,\leq\,\frac{1}{2}\,\log_{1/(r_{s}+\delta)}\,\,\rho_{\,k,s}$.
Taking the limit as $\delta\to 0$, we complete the proof.

$\Box$

##  6 The tile B-splines

As an example of application of Theorem 5.7, we compute the regularity of the
tile B-splines. We are going to show that, unless the dilation matrix $M$ can
be factored to a block-diagonal form by means of an integer basis, the Hölder
exponent is computed by a simple formula
$\alpha(\varphi)\,=\,\frac{1}{2}\,\log_{1/r}\,\rho_{\ell}$, where $\ell$ is
the order of the B-spline. Thus, the exponent of regularity is computed
without involving zeros from the set $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\,\bigr{\\}}$. The proof is
based on two arguments: geometrical (tile cylinders) and combinatorial
(rational subspaces in ${\mathbb{R}}^{n}$). First we introduce tile cylinders
and prove their basic properties. We show that if the assumptions of Theorem
5.8 are not satisfied, the tiling by translates of $G$ contains a nontrivial
cylinder parallel to an eigenspace of $M$. Then we prove that this eigenspace
is a rational subspace of ${\mathbb{R}}^{n}$. This means that if $M$ does not
have rational eigenspaces, then the assumptions of Theorem 5.8 are satisfied.

6.1. Tile cylinders and rational subspaces in ${\mathbb{R}}^{n}$

A cylinder parallel to a subspace $J\subset{\mathbb{R}}^{n}$ is a closed
nonempty subset of ${\mathbb{R}}^{n}$ invariant with respect to all
translations parallel to $J$. In particular, if $J={\mathbb{R}}^{n}$, then
$C={\mathbb{R}}^{n}$; if $J=\\{{\boldsymbol{0}}\\}$, then $C$ is an arbitrary
closed set. We usually avoid those cases and assume that $J$ is a proper
nontrivial subspace of ${\mathbb{R}}^{n}$.

We say that a given tiling
$\\{G+{\boldsymbol{k}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$ contains a
cylinder $C$ if there is a subset $E\subset{\mathbb{Z}}^{n}$ such that
$C=\bigcup_{{\boldsymbol{k}}\in E}(G+{\boldsymbol{k}})$. A cylinder contained
in a given tiling will be referred to as a tile cylinder. The set of all tile
cylinders parallel to a subspace $J$ is closed with respect to the operations
of intersection, union, and difference (the latter two operations are followed
by taking the closure). Every tiling possesses at least one cylinder parallel
to $J$, which is $C={\mathbb{R}}^{n}$. This case is trivial. All other tile
cylinders, if they exist, are called proper.

A tile cylinder is minimal if it does not contain another tile cylinder. One
can also specify a subspace $J$ and consider the minimal tile cylinder among
those parallel to $J$.

###### Lemma 6.1

For an arbitrary tiling and a subspace $J\subset{\mathbb{R}}^{n}$, there
exists a minimal tile cylinder parallel to $J$.

Proof. Choose a tile $G$ and consider the intersection $C$ of all cylinder
parallel to $J$ and containing $G$. This cylinder may not be minimal since it
can include another cylinder $C^{\prime}$ which does not contain $G$. In this
case, however, the closure of the set $C\setminus C^{\prime}$ is a cylinder
containing $G$, which contradicts the definition of $C$. Hence, $C$ is
minimal.

$\Box$

###### Lemma 6.2

All minimal tile cylinders parallel to $J$ are integer translates of each
other. Every tile cylinder parallel to $J$ is partitioned to the minimal ones.

Proof. The intersection of all tile cylinders parallel to $J$ and containing
the tile $G$ is a minimal cylinder (see the proof of Lemma 6.1 above). Denote
it by $C$. Any other minimal cylinder $C^{\prime}$ contains some tile
$G^{\prime}=G+{\boldsymbol{k}}$. Hence, the cylinder
$C^{\prime}-{\boldsymbol{k}}$ contains $G$ and therefore, it contains $C$,
because $C$ is minimal. If $C$ is a proper subset of
$C^{\prime}-{\boldsymbol{k}}$, then the cylinder $C+{\boldsymbol{k}}$, which
contains $G+{\boldsymbol{k}}=G^{\prime}$, is a proper subset of $C^{\prime}$.
This is impossible since $C^{\prime}$ is minimal. Thus,
$C=C^{\prime}-{\boldsymbol{k}}$, and we proved that all minimal cylinders are
integer translates of each other. Now we introduce an equivalence relation on
the set of tiles: two tiles are equivalent if they belong to one minimal
cylinder. If $C_{0}$ is an arbitrary tile cylinder parallel to $J$, then we
take the maximal system of non-equivalent tiles in it. Then the corresponding
minimal cylinders fill $C_{0}$.

$\Box$

Since ${\mathbb{R}}^{n}$ is also a cylinder, we obtain

###### Corollary 6.3

For every tiling, the minimal tile cylinders form a partition of the space
${\mathbb{R}}^{n}$.

A linear subspace $L\subset{\mathbb{R}}^{n}$ of dimension $d$ is called
rational if it is spanned by several integer vectors. It is known that every
rational subspace has a basis of integer vectors that spans the whole integer
part of $L$, which is $L\cap{\mathbb{Z}}^{n}$, by integer linear combinations.
It will be referred to as an elementary basis. The characteristic property of
the elementary basis is that its Gram matrix has the determinant one. Indeed,
this condition means that the parallelepiped spanned by the vectors of this
basis has the $d$-dimensional volume one, and hence, it spans the integer
lattice on $L$. Every elementary basis can be complemented to an elementary
basis of ${\mathbb{R}}^{n}$, which spans the lattice ${\mathbb{Z}}^{n}$. There
is an integer matrix that maps an elementary basis to the first $d$ vectors of
the canonical basis of ${\mathbb{R}}^{n}$.

The linear span and the intersection of several rational subspaces are also
rational subspaces. This implies that every subspace
$J\subset{\mathbb{R}}^{n}$ possesses the minimal by inclusion rational
subspace $L$ containing $J$. This subspace will be called the minimal rational
subspace of $J$. There are countably many proper rational subspaces of
${\mathbb{R}}^{n}$. For every vector out of those subspaces, its minimal
rational subspace is the entire ${\mathbb{R}}^{n}$. Thus, for almost all (in
the Lebesgue measure) vectors of ${\mathbb{R}}^{n}$, the minimal rational
subspace coincides with ${\mathbb{R}}^{n}$.

6.2. The polynomial cones for
the tile B-splines

We consider a tile B-spline $\varphi$ of order $\ell$, which is a convolution
of $\ell+1$ functions $\chi_{G}$. For this tile B-spline,
$\widehat{\varphi}=\widehat{\chi_{G}}^{\ell+1}$, therefore $\mbox{\em
nill}\,_{k}(\widehat{\varphi})=\mbox{\em nill}\,(\widehat{\varphi})$ for all
$k\leq\ell$. For an arbitrary $k\leq\ell$ and for every $s=1,\ldots,q$, we
consider the space $P_{k,s}$ of polynomials from ${\mathcal{P}}_{\Omega}$ that
have zeros of order at least $2(k+1)$ on the set $\pi\\{\mbox{\em
nill}\,(\widehat{\varphi})\,,\,J_{s}^{\perp}\\}$ and the space
$P_{k}={\mathcal{P}}^{(k)}$ of polynomials from ${\mathcal{P}}_{\Omega}$ that
have zero of order at least $2(k+1)$ at the origin. The following theorem
claims that for a tile B-spline, each subspace $P_{k,s}$ coincides with
$P_{k}$, unless the corresponding tiling has a proper cylinder parallel to the
minimal rational subspace $L_{s}$ of $J_{s}$.

###### Theorem 6.4

Let $\varphi$ be the tile B-spline of order $\ell$ generated by a tile $G$;
then for every $k\leq\ell$ and for every $s=1,\ldots,q$ , the following holds:
either $P_{k,s}=P_{k}$ or the tiling
$\\{G+{\boldsymbol{k}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$ contains a
proper cylinder parallel to the minimal rational subspace of $J_{s}$.

The proof of Theorem 6.4 begins with several auxiliary results. For a rational
subspace $L\subset{\mathbb{R}}^{n}$, we say that its subset $A\subset L$ is
dense in $L/{\mathbb{Z}}^{n}$ if the union of the translates
$A+{\boldsymbol{k}},\ {\boldsymbol{k}}\in{\mathbb{Z}}^{n}\cap L$ is dense in
$L$. The following fact is well-known:

###### Lemma 6.5

If $J\subset{\mathbb{R}}^{n}$ is a subspace and $L$ is its minimal rational
subspace, then $J$ is dense in $L/{\mathbb{Z}}^{n}$.

###### Proposition 6.6

The minimal tile cylinder parallel to a subspace $J$ is parallel to its
minimal rational subspace.

Proof. Let $C$ be a tile cylinder and $G\subset C$ be one of its tiles,
$C=\bigcup_{{\boldsymbol{k}}\in
E}(G+{\boldsymbol{k}}),\,E\subset{\mathbb{Z}}^{n}$. For a given subspace
$V\subset{\mathbb{R}}^{n}$, we denote by $G_{V}$ the intersection of $G$ with
the set $\bigcup_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}(V+{\boldsymbol{k}})$
and by $G_{V}^{\prime}$ the intersection of $G$ with the set
$\bigcup_{{\boldsymbol{k}}\in E}(V+{\boldsymbol{k}})$. Since the cylinder $C$
is parallel to $J$, it follows that $G_{J}^{\prime}=G_{J}$. Furthermore, if
$L$ is the minimal rational subspace for $J$, then $G_{J}$ is dense in
$G_{L}$. Hence, $G_{J}^{\prime}$ is dense in $G_{L}$. Consequently,
$G_{L}^{\prime}$ is dense in $G_{L}$ and by compactness,
$G_{L}^{\prime}=G_{L}$. Therefore, $C$ is parallel to $L$.

$\Box$

Proposition 6.6 allows us to consider only rational subspaces parallel to
cylinders. Since the linear span of several subspaces parallel to a cylinder
is also parallel to it, there is the maximal by inclusion subspace parallel to
the cylinder called the generatrix. Proposition 6.6 yields that the generatrix
of a minimal tile cylinder $C$ contains all (not necessarily rational)
subspaces parallel to $C$.

###### Corollary 6.7

The generatrix of the minimal tile cylinder parallel to a subspace $J$ is a
rational subspace.

Proof of Theorem 6.4. As the first step, we reduce the theorem to the case
$k=0$, i.e., to the tile function $\phi=\chi_{G}$. Indeed, in view of
Proposition 5.2, for the tile B-spline $\varphi$, either $P_{k,s}=P_{k}$ or
there exists a non-integer vector ${\boldsymbol{z}}\in J_{s}^{\perp}$ such
that for every integer ${\boldsymbol{k}}$, the vector
${\boldsymbol{z}}+{\boldsymbol{k}}$ is either orthogonal to $J_{s}$ or is a
zero of the function $\widehat{\varphi}$ of order at least $k+1$. Since
$\widehat{\varphi}\,=\,\bigl{(}\,{\widehat{\phi}}\,\bigr{)}^{\,\ell}$, it
follows that the functions $\widehat{\varphi}$ and $\widehat{\phi}$, have the
same sets of zeros. So, that condition is equivalent to
$\widehat{\phi}({\boldsymbol{z}}+{\boldsymbol{k}})=0$ for all
${\boldsymbol{k}}\in{\mathbb{Z}}^{n}$ unless
${\boldsymbol{z}}+{\boldsymbol{k}}\in J_{s}^{\perp}$. Thus, one needs to prove
the theorem for the case $k=0$, i.e., $\varphi=\phi$.

For every ${\boldsymbol{h}}\in J_{s}$, consider the generalized derivative
$\phi_{{\boldsymbol{h}}}\in{\mathcal{S}}^{\prime}$ (the generalized derivative
along the vector ${\boldsymbol{h}}$). We have

$\widehat{\phi_{{\boldsymbol{h}}}}({\boldsymbol{z}}+{\boldsymbol{k}})\ =\ 2\pi
i\,({\boldsymbol{h}},{\boldsymbol{z}}+{\boldsymbol{k}})\,\widehat{\phi}({\boldsymbol{z}}+{\boldsymbol{k}})\
=\ 0\qquad\mbox{{\rm for all}}\qquad{\boldsymbol{k}}\in{\mathbb{Z}}^{n}.$

Indeed, for ${\boldsymbol{k}}\neq{\boldsymbol{0}}$, we have
$\widehat{\phi}({\boldsymbol{z}}+{\boldsymbol{k}})\,=\,0$, and for
${\boldsymbol{k}}={\boldsymbol{0}}$, we have
$({\boldsymbol{h}},{\boldsymbol{z}}+{\boldsymbol{k}})\,=\,({\boldsymbol{h}},{\boldsymbol{z}})\,=\,0$.
Consequently, by the Poisson summation formula,
$\sum\limits_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}e^{2\pi
i({\boldsymbol{z}},{\boldsymbol{q}})}\phi_{{\boldsymbol{h}}}(\mathord{\,\cdot\,}+{\boldsymbol{q}})\,=\,0$.
Consider the function

$F({\boldsymbol{x}})\ =\ \sum_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}\,e^{2\pi
i({\boldsymbol{z}},{\boldsymbol{q}})}\,\phi({\boldsymbol{x}}+{\boldsymbol{q}}).$
(14)

The summation is well-defined because $\phi$ is compactly-supported. In fact,
$F$ is a piecewise-constant function corresponding to the tiling
$\\{G+{\boldsymbol{k}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$. Moreover,
since the set of coefficients $\\{e^{2\pi
i({\boldsymbol{z}},{\boldsymbol{q}})}\\}_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}$
is bounded, it follows that the series converges also in
${\mathcal{S}}^{\prime}$ and can be termwise differentiated along the
direction ${\boldsymbol{h}}$. Therefore, $F_{{\boldsymbol{h}}}=0$ and $F$ is
an identical constant along every line parallel to ${\boldsymbol{h}}$. Since
${\boldsymbol{h}}$ is an arbitrary vector from $J_{s}$, it follows that $F$ is
an identical constant along every affine plane with the linear part $J_{s}$.
On the other hand, $F$ is a piecewise-constant function on the tiling, hence,
all level sets of $F$ are either empty or cylinders parallel to $J_{s}$.
Moreover, the vector ${\boldsymbol{z}}$ is non-integer, consequently, the
coefficients $e^{2\pi
i({\boldsymbol{z}},{\boldsymbol{q}})},\,{\boldsymbol{q}}\in{\mathbb{Z}}^{n}$,
are not all equal. That is why the level sets of $F$ are proper cylinders. Now
the theorem follows from Proposition 6.6.

Conversely, if the tiling contains a proper cylinder parallel to $L_{s}$, then
it is parallel to $J_{s}$. Therefore, ${\mathbb{R}}^{n}$ is split to equal
tile cylinders. Associating an arbitrary number to each of those cylinders (we
choose all those numbers
$\\{c_{{\boldsymbol{q}}}\\}_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}$ to be
different) we obtain a piecewise-constant function
$F(\mathord{\,\cdot\,})\,=\,\sum_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}c_{{\boldsymbol{q}}}\,\phi(\cdot+{\boldsymbol{q}})$
which is constant on every cylinder. This implies that $F$ is constant along
any line parallel to $J_{s}$, which means that for every ${\boldsymbol{h}}\in
J_{s}$, the generalized derivative $F_{{\boldsymbol{h}}}$ is zero. Thus,
$\sum_{{\boldsymbol{q}}\in{\mathbb{Z}}^{n}}c_{{\boldsymbol{q}}}\,\phi_{{\boldsymbol{h}}}(\cdot+{\boldsymbol{q}})\,=\,0$.
Therefore, $c_{{\boldsymbol{q}}}=e^{2\pi
i({\boldsymbol{z}},{\boldsymbol{q}})}$ for some
${\boldsymbol{z}}\in{\mathbb{R}}^{n}$ (see [44]). Since $F$ is not an
identical constant, it follows that ${\boldsymbol{z}}\notin{\mathbb{Z}}^{n}$.
Now by the Poisson summation formula it follows that
$\,\widehat{\phi_{{\boldsymbol{h}}}}({\boldsymbol{z}}+{\boldsymbol{k}})\,=\,0,\
{\boldsymbol{k}}\in{\mathbb{Z}}^{n}$. Thus, $2\pi
i\,({\boldsymbol{h}},{\boldsymbol{z}}+{\boldsymbol{k}})\,\widehat{\phi}({\boldsymbol{z}}+{\boldsymbol{k}})=0,\
{\boldsymbol{k}}\in{\mathbb{Z}}^{n}$. This is equivalent to say that
${\boldsymbol{z}}\in\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\phi})\,,\,J_{s}^{\perp}\bigr{\\}}$. Since $\mbox{\em
nill}\,(\widehat{\varphi})=\mbox{\em nill}\,(\widehat{\phi})$, we see that the
set $\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$ contains a non-integer
point, hence, $P_{k,s}\neq P_{k}$.

$\Box$

6.3. The regularity of tile B-splines

Now we are formulating the main result of this section.

###### Theorem 6.8

For every tile B-spline $\varphi$, we have the alternative: either
${\alpha(\varphi)\,=\,\frac{1}{2}\log_{1/r}\,\rho_{\,\ell}}$, or the
corresponding tiling
$\\{G+{\boldsymbol{k}}\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$ contains a
proper cylinder parallel to the minimal rational subspace $L_{1}$ of $J_{1}$.
In the latter case, the following hold:

1) $L_{1}$ is a proper subspace of ${\mathbb{R}}^{n}$ invariant with respect
to $M$;

2) In a suitable integer basis of ${\mathbb{R}}^{n}$, the matrix $M$ becomes
an integer upper block-triangular matrix with the first block $A$
corresponding to $L_{1}$:

$M\ =\ \left(\begin{array}[]{cc}A&*\\\ 0&B\end{array}\right)\,,$ (15)

Proof. If $\alpha(\varphi)\neq\frac{1}{2}\log_{1/r}\,\rho_{\,\ell}$, then, by
Theorem 5.3, $P_{1}$ is a proper subspace of $P$, which, in view of Theorem
6.4, implies that $L_{1}$ is a proper subspace of ${\mathbb{R}}^{n}$. Since
$J_{1}\subset L_{1}$ and $MJ_{1}=J_{1}$, it follows that the rational subspace
$ML_{1}$ also contains $J_{1}$ and hence contains $L_{1}$. Thus, $L_{1}\subset
ML_{1}$ and therefore, $L_{1}=ML_{1}$ since the matrix $M$ is nondegenerate.
Choosing an elementary basis of $L_{1}$ and complementing it to an elementary
basis in ${\mathbb{R}}^{n}$, we obtain a basis in which $M$ becomes an integer
matrix of the form (15).

$\Box$

###### Theorem 6.9

For a tile B-spline, the inequality
$\alpha(\varphi)\,\neq\,\frac{1}{2}\log_{1/r}\,\rho_{\,\ell}$ is possible only
if $m\geq 4$; in the bivariate case ($n=2$), it is possible only if $m\geq 6$.

Proof. By Theorem 6.8, the inequality
$\alpha\neq\frac{1}{2}\log_{1/r}\,\rho_{\ell}$ implies that in a suitable
elementary basis, the matrix $M$ gets the upper block-triangular form (15)
with integer entries, where the subspace $L_{1}$ corresponds to the block $A$.
Since $J_{1}\subset L_{1}$, it follows that $\rho\,=\,\rho(A)>\rho(B)$. On the
other hand, the matrices $A$ and $B$ are both expanding and integer, hence,
their determinants are at least two in absolute value. Therefore, $m\,=\,|{\rm
det}\,M|\,\geq\,4$. In the two-dimensional case, $A$ and $B$ are numbers and
$|A|>|B|\geq 2$, so $|A|\geq 3$, and $m=|AB|\geq 6$.

$\Box$

###### Remark 6.10

Both of the lower bounds given by Theorem 6.9 are sharp. For bivariate
splines, the $6$-digit case is illustrated in Example 7.1 The three-
dimensional $4$-digit case is given by “bear cylinders” in Example 7.2.

##  7 Examples and numerical results. The super-smoothness

To compute the $L_{2}$-regularity of a refinable function by Theorems 5.7 and
5.8 one needs to write the matrix of the operator ${\mathcal{T}}$, find its
restrictions to the invariant subspaces $P_{k,s}$, compute their spectral
radii $\rho_{k,s}$ (i.e., the Krein-Rutman leading eigenvalues) and substitute
those values in formula (11). We obtain the biggest $k$ for which $\varphi\in
W_{2}^{k}$ and the Hölder exponent of the $k$th derivative. The technical
details are given in Appendix. We begin with two simple examples that prove
the sharpness of Theorem 6.9. In the sequel of this section we use the same
notation $M,D$, etc., in different examples.

7.1. Two examples for Theorem 6.9

If $n\geq 3$, then the equality
$\alpha(\varphi)\,=\,\frac{1}{2}\log_{1/r}\,\rho_{\,\ell}$ may fail for
$4$-digit tile B-splines; if $n=2$, then it may for $6$-digit ones.

###### Example 7.1

Consider tile B-splines $\varphi_{\ell}$ of order $\ell$ generated by the same
diagonal dilation matrix:

$M\ =\begin{pmatrix}3&0\\\ 0&2\end{pmatrix}$

and by different digit systems $D$. We have $r_{1}=3,r_{2}=2$, so $r=r_{1}=3$,
and $m=6$. For the system $D\,=\,\bigl{\\{}\,(i,j)\,:\ i\in\\{0,1,2\\},\
j\in\\{0,1\\}\,\\}$, the corresponding tile is a unit square, therefore,
$\varphi_{\ell}=B_{\ell}$ and hence,
$\,\alpha(\varphi_{\ell})=\ell+\frac{1}{2}$ for all $\ell\geq 0$. On the other
hand, the corresponding refinement equation is a direct product of two
univariate refinement equations, hence,
$\rho_{0,1}=\frac{1}{3},\rho_{0,2}=\frac{1}{2}$, so $\rho_{\,0}=\frac{1}{2}$.
For $\ell=0$, we have $\alpha=0.5$, while
$\,\frac{1}{2}\log_{1/r}\,\rho_{\,0}\,=\,\frac{1}{2}\log_{1/3}\,\frac{1}{2}\,=\,0.315...$.
The same situation takes place for all $\ell\geq 1$:
$\alpha=\ell+\frac{1}{2}\,>\,\frac{1}{2}\log_{1/r}\,\rho_{\,\ell}\,=\,\frac{1}{2}\log_{1/3}\,2^{-(2\ell+1)}\,=\,\bigl{(}\ell+\frac{1}{2}\bigr{)}\,\log_{3}2$.
This anysotropic equation is a direct product of isotropic ones; such examples
are well-known. The following example with the same matrix $M$ is less
trivial:

Figure 1: 6-digit tile, Example 7.1

Figure 2: 4-digit bear cylinder, Example 7.2.

In the digit set $D$, we replace the digit $(0,1)$ by $(3,1)$. The new set $D$
is not a direct product and neither the corresponding refinement equation. For
$\ell=0$, when $\varphi_{\ell}=\varphi$ is the indicator function of the tile
$G$ (fig. 2), we again have $\rho_{0,1}=\frac{1}{3},\rho_{0,2}=\frac{1}{2}$
and $\alpha=\frac{1}{2}$. For the higher order $\ell$, we also have
$\alpha\,>\,\frac{1}{2}\log_{1/r}\,\rho_{\ell}$ as its is seen below:

For $\ell=1$, $\ \alpha=1.630...\quad$ while
$\quad\frac{1}{2}\log_{1/3}\,\rho_{1}\,=\,1.028...$;

For $\ell=2$, $\ \alpha=2.261...\quad$ while
$\quad\frac{1}{2}\log_{1/3}\,\rho_{2}\,=\,1.426...$;

For $\ell=3$, $\ \alpha=2.892...\quad$ while
$\quad\frac{1}{2}\log_{1/3}\,\rho_{3}\,=\,1.824...;$.

Theorem 6.8 guarantees the existence of the tile cylinder in this example.
Indeed, this is the horizontal strip $\Pi\,=\,\bigl{\\{}(x_{1},x_{2})\,:\
0\leq x_{2}\leq 1\,\bigr{\\}}$ formed by the translates
$G+(k,0)\,,k\in{\mathbb{Z}}$.

###### Example 7.2

If $n\geq 3$, then Theorem 6.9 asserts that
$\alpha\,=\,\frac{1}{2}\log_{1/r}\rho_{\,\ell}$, unless $m\geq 4$. The example
for $m=4$ is given by the so-called Bear cylinder in ${\mathbb{R}}^{3}$. The
base of this cylinder is Bear tile $G_{b}$ generated in ${\mathbb{R}}^{2}$ by
the isotropic dilation matrix:

$M_{b}\ =\ \begin{pmatrix}1&-2\\\ 1&\quad 0\end{pmatrix}$ (16)

and digits $\,D\,=\,\bigl{\\{}\,(0,0),\,(1,0)\,\bigr{\\}}$. Here
$m=2,r=\sqrt{2}$ and $\rho_{0}=0.7607...$, therefore,
$\alpha(\varphi^{(b)})=0.3946...$, where $\varphi^{(b)}$ is the indicator
function of $G_{b}$. The Bear cylinder $G\,=\,[0,1]\,\times\,G_{b}$ satisfies
a refinement equation with the dilation $3\times 3$-matrix $M$ that consists
of two diagonal blocks: $\\{2\\}$ and $M_{b}$. Thus, $r_{1}=2,r_{2}=\sqrt{2}$,
hence, $r=2$. The set of digits is $(0,0,0),(0,1,0),(1,0,0),(1,1,0)$. We have
$\alpha(\varphi)=\alpha(\varphi^{(b)})=0.3946...$, while
$\frac{1}{2}\log_{1/r}\rho_{\,0}\,=\,0.1973$.

7.2. The super-smooth tile B-splines

We illustrate our method by computing the higher order regularity of tile
B-splines for different dilation matrices $M$. In particular, we find several
families of tile B-splines $\varphi_{\ell}$, whose regularity exceeds that of
the cardinal B-splines $B_{\ell}$, i.e.,
$\alpha(\varphi)\,>\,\ell+\frac{1}{2}$. This phenomenon is rather surprising,
we call it super-smoothness. Three families of tile B-splines are found in
${\mathbb{R}}^{2}$: one of them is two-digit, i.e., $|{\rm det}\,M|=2$ (the
Bear B-splines), two others are three-digit (see fig. 3). The results are
shown in Tables 1, 2, and 3. Table 1 contains the Hölder exponents in $L_{2}$
for the three “super-smooth” tile B-splines (lines 2 – 4).

Figure 3: Tiles generating super-smooth B-splines: the Bear and the three-
digit tiles ThreeDig1, ThreeDig2. For each tile, the self-similar partition
$G=\cup_{{\boldsymbol{d}}\in D}M^{-1}(G+{\boldsymbol{d}})$ is shown

The line 2 shows the Hölder exponents of the Bear B-splines generated by the
Bear tile (see Example 7.2). This is one of three existing two-digit tiles on
the plane, along with the Square and the Dragon [3, 52]. It possesses many
remarkable properties, see [53] for an overview. In particular, their
B-splines $\varphi^{(b)}_{\ell}$ possesses the super-smoothness in
$C({\mathbb{R}})$ at least for $\ell=2,3,4$ and are useful for construction of
wavelets and subdivision schemes [51]. By the new method it is possible to
find the regularity in $L_{2}({\mathbb{R}})$ for higher orders $\ell$, Table 1
demonstrates them for $\ell\leq 8$. We see that
$\alpha(\varphi_{\ell})\,>\,\alpha(B_{\ell})$ for all $\ell\geq 1$.

The same super-smoothness phenomenon is observed for two three-digit tiles,
which are referred to as ThreeDig1 and ThreeDig2. They are generated by the
dilation matrices $M_{1},M_{2}$ given in (17) and by the same digit set
$D\,=\,\bigl{\\{}(0,0)\,,\,(1,0)\,,\,(0,1)\,\bigr{\\}}$:

$M_{1}\ =\ \begin{pmatrix}1&-2\\\ 1&\quad 1\end{pmatrix}\ ;\qquad M_{2}\ =\
\begin{pmatrix}1&-1\\\ 1&\quad 2\end{pmatrix}$ (17)

The results for the B-splines $\varphi_{\ell}^{(2)}$ generated by ThreeDig2
are especially interesting, since their regularity are very close to the
maximal possible level. It is well-known that if for a stable refinable
function $\varphi_{\ell}$, the order of sum rules is equal to $\ell$, then it
cannot belong to $W^{\ell+1}_{2}$, i.e., $\alpha(\varphi_{\ell})<\ell+1$. We
see that for $\varphi_{\ell}^{(2)}$ this critical level is almost achieved
already for $\ell=4$, when $\alpha(\varphi_{4})>4.9861$. For $\ell=8$, we have
$\alpha(\varphi_{8})>8.9997$.

$\ell$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---
$B_{\ell}$ | 0.5 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5
Bear | 0.3946 | 1.5372 | 2.6323 | 3.7063 | 4.7669 | 5.8173 | 6.859 | 7.893 | 8.9199
ThreeDig1 | 0.3116 | 1.4913 | 2.6078 | 3.6918 | 4.7563 | 5.8076 | 6.8488 | 7.8819 | 8.9084
ThreeDig2 | 0.3691 | 1.6571 | 2.8536 | 3.9518 | 4.9861 | 5.9961 | 6.9989 | 7.9997 | 8.9997
Table 1: The Hölder exponents of the super-smooth bivariate tile B-splines:
the Bear, ThreeDig1, and ThreeDig2

The Dragon, which is the third two-digit plane tile, generates B-splines with
the regularity less than the standard cardinal B-splines.

$\ell$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
Dragon | 0.2382 | 1.0962 | 1.8038 | 2.4395 | 3.0562 | 3.6688
ThreeDig3 | 0.1712 | 0.9805 | 1.5872 | 2.1363 | 2.6743 | 3.21
Table 2: The Hölder exponents of bivariate tile B-splines of lower regularity

Among the three-digit plane tiles (there are infinitely many of them) we have
found only two with super-smoothness. For others we have
$\alpha<\ell+\frac{1}{2}$. Table 2 shows the regularity of one of those
examples (we called it ThreeDig3), which are still lower than for the Dragon
tile.

We are not aware of any super-smooth tile B-splines in ${\mathbb{R}}^{3}$. At
least, there is no such examples in the two-digit case. In fact, there exist
precisely seven two-digit tiles in ${\mathbb{R}}^{3}$ [3], we did the
computations for all of them. The first one is the cube producing the cardinal
B-spline $B_{\ell}$, the six other produce tile B-splines of regularity lower
thar $B_{\ell}$. Table 3 shows the results for two of them (see the
classification in [53]).

$\ell$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
Cube | 0.5 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5
3d Type 1 | 0.2328 | 1.3299 | 2.2125 | 2.9545 | 3.6933 | 4.4319
3d Type 2 | 0.1173 | 0.6236 | 0.9755 | 1.3035 | 1.6296 | 1.9556
Table 3: The $L_{2}$-regularity of two-digit tile B-splines in
${\mathbb{R}}^{3}$

###### Remark 7.3

The super-smoothness of tile B-splines may seem contradictory: the
autoconvolution of a square is less smooth than the autoconvolution (of the
same order) of an irregular fractal-like tile. We are not aware of any
explanation of this phenomenon. Nevertheless, one argument can be given: a
convolution of a polygon with itself cannot be from $C^{1}$, while for some
other figures it can. The details are in fig. 4. Let $f_{1}$ be the indicator
of the square $\\{(x,y)\,:\,|x|\leq 1,|y|\leq 1\\}$ and $f_{2}$ be that of the
unit disc centered at the origin, $F_{i}=f_{i}*f_{i},\,i=1,2$. Then $F_{1}$ is
not differentiable at ${\boldsymbol{x}}=(1,0)$, while $F_{2}$ is. Indeed,
$F_{1}(1-t,0)$ is equal to the area of shadowed rectangle, which is $2t$ if
$t>0$ and $0$ otherwise. On the other hand, it is easily shown that
$F_{2}({\boldsymbol{x}}+{\boldsymbol{h}})=O(\|{\boldsymbol{h}}\|^{3/2})$,
hence $F_{2}^{\prime}({\boldsymbol{x}})=0$.

Figure 4: The different smoothness of autoconvolutions of a square and of a
disc

7.3. The numerical complexity

The efficiency of the Littlewood-Paley technique in comparison to the matrix
approach is explained by the dimension of the Perron eigenvalue problem. In
the univariate case, the matrix approach deals with the matrix size of order
$\,\frac{1}{2}\,N^{2}$, while the Littlewood-Paley method does with the size
of order $2N$, where $N$ is the degree of the mask. For the multivariate case,
the sizes of both matrices are bigger, but the size in the Littlewood-Paley
method (i.e., the dimension of the space ${P_{\ell}}$) is still much less than
one of the matrix method. Table 4 compares these sizes for the Bear B-splines.
The situation in other numerical examples is similar.

$\ell$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
Littlewood-Paley | 4 | 11 | 20 | 55 | 76 | 101 | 162 | 203 | 242 | 333
Matrix approach | 9 | 81 | 225 | 324 | 1089 | 1764 | 3969 | 5329 | 5041 | 9409
$\alpha(\varphi_{\ell}^{(b)})$ | 0.395 | 1.537 | 2.632 | 3.706 | 4.767 | 5.817 | 6.86 | 7.893 | 8.92 | 9.94
Table 4: The matrix sizes in the two methods: computation for the Bear
B-splines.

##  8 Optimality of the tile B-splines

We establish the following optimal property of tile B-splines: they possess
the smallest number of nonzero coefficients among all refinable functions of a
given order of approximation.

For the sake of simplicity, we restrict to the case $m=2$, i.e., to two-digit
refinable functions. Thus, among all refinable functions corresponding to a
given dilation matrix $M$ with $|{\rm det}\,M|=2$ and generating (by integer
shifts) the space of algebraic polynomials of a given degree, the tile
B-spline has the minimal number $|Q|$ of nonzero coefficients. This property,
which will be referred to as the minimal support of the mask, also holds for
tile B-splines in the following classes of refinable functions:

1) of a given rate of approximation of smooth functions by contractions of
integer translates;

This implies that among all refinable functions generating the multiresolution
analysis (MRA) $\\{V_{j}\\}_{j\in{\mathbb{Z}}}$ with a given rate of
approximation by the subspaces $V_{j}$, the B-splines have the minimal support
of the mask. Consequently, they have the fastest algorithm of computation of
the wavelet coefficients produced by the corresponding MRA.

2) of a given degree of convergence of the subdivision scheme.

This means, in particular, that the subdivision scheme generated by the tile
B-spline has the least algorithmic complexity (the number of operations to
realize one iteration) among all schemes reproducing the space of polynomials
of a given degree.

The digit set for the matrix $M$ is
$D\,=\,\\{{\boldsymbol{0}},{\boldsymbol{d}}\\}$, where
${\boldsymbol{d}}\in{\mathbb{Z}}^{n}\setminus M{\mathbb{Z}}^{n}$. The digit
set for the matrix $M^{T}$ is
$D^{*}\,=\,\\{{\boldsymbol{0}},{\boldsymbol{d}}^{*}\\}$, where
${\boldsymbol{d}}^{*}\in{\mathbb{Z}}^{n}\setminus M^{T}{\mathbb{Z}}^{n}$.

The following theorem is well-known (see, for instance, [30]).

Theorem B. For a given integer $\ell\geq 0$, the following properties of a
stable two-digit refinable function are equivalent:

1) the integer translates
$\\{\varphi(\mathord{\,\cdot\,}+{\boldsymbol{k}})\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$
generate the space of algebraic polynomials of degree $\ell$;

2) the refinement equation satisfies the sum rules of order $\ell$, i.e., the
mask ${\boldsymbol{c}}({\boldsymbol{\xi}})$ has zero of order $\,\geq\,\ell+1$
at the point ${\boldsymbol{\xi}}=M^{-T}{\boldsymbol{d}}^{*}$.

3) the function $\varphi$ has the rate of approximation at least $\ell+1$,
i.e., the distance from an arbitrary function $f\in
C^{\ell+1}({\mathbb{R}}^{n})$ to the space generated by the functions
$\\{\varphi(N\mathord{\,\cdot\,}+{\boldsymbol{k}})\\}_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n}}$
is $O(N^{-\ell-1})$.

We denote by $\nu(\varphi)$ the accuracy of the refinable function $\varphi$,
i.e., the maximal $\ell\geq 0$ for which the conditions 1) - 3) from Theorem B
are satisfied. This is known that if $\varphi\in W^{\ell}_{2}$, then
$\nu(\varphi)\geq\ell$ [30].

###### Theorem 8.1

If a two-digit refinable function $\varphi$ satisfies $\nu(\varphi)\geq\ell$,
then its refinement equation possesses at least $\ell+2$ nonzero coefficients.

The mask of the tile B-spline
${\boldsymbol{c}}({\boldsymbol{\xi}})\,=\,\bigl{(}\frac{e^{2\pi
i\,({\boldsymbol{d}},{\boldsymbol{\xi}})}+1}{2}\bigr{)}^{\ell+1}$ has exactly
$\ell+2$ nonzero coefficients. Therefore, for each dilation matrix $M$ such
that $|{\rm det}\,M|=2$, among all refinable functions with
$\nu(\varphi)\geq\ell$, the tile B-spline of order $\ell$ has the minimal
support of the mask $|Q|=\ell+2$.

Proof of Theorem 8.1. Consider an arbitrary refinement equation with a mask
${\boldsymbol{c}}(\mathord{\,\cdot\,})$ satisfying the sum rules of order
$\ell$. Denote ${\boldsymbol{v}}=M^{-T}{\boldsymbol{d}}^{*}$ and define the
univariate polynomial
${\boldsymbol{q}}(t)={\boldsymbol{c}}\,(2t\,{\boldsymbol{v}}),\
t\in{\mathbb{R}}$. This polynomial has a root of multiplicity at least
$\ell+1$ at the point $t=\frac{1}{2}$. Since $|{\rm det}\,M|=2$, it follows
that $2\,M^{-T}$ is an integer matrix, and therefore,
$2{\boldsymbol{v}}\in{\mathbb{Z}}^{n}$. This implies that the scalar products
$(2{\boldsymbol{v}},{\boldsymbol{k}})$ are integer for all
${\boldsymbol{k}}\in{\mathbb{Z}}^{n}$ and hence,

${\boldsymbol{q}}(t)\ =\
\frac{1}{2}\,\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}}\,c_{{\boldsymbol{k}}}\,e^{2\pi
i\,(2t{\boldsymbol{v}},{\boldsymbol{k}})}\ =\
\sum_{s\in{\mathbb{Z}}}\left(\,\sum_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n},\,(2{\boldsymbol{v}},{\boldsymbol{k}})=s}\,\frac{1}{2}\,c_{{\boldsymbol{k}}}\,\right)\,e^{2\pi
i\,st}\ =\ \sum_{s\in{\mathbb{Z}}}\,q_{s}\,e^{2\pi i\,st},\ $

where $\
q_{s}\,=\,\frac{1}{2}\,\sum\limits_{{\boldsymbol{k}}\in{\mathbb{Z}}^{n},(2{\boldsymbol{v}},{\boldsymbol{k}})=s}\,c_{{\boldsymbol{k}}}$.
Since ${\boldsymbol{q}}^{(r)}\bigl{(}\frac{1}{2}\bigr{)}\,=\,0$ for all
$r=0,\ldots,\ell$, we have

$\sum_{s\in{\mathbb{Z}}}\,q_{s}\bigl{(}2\pi is\bigr{)}^{r}e^{\pi
is}\quad=\quad 0,\qquad r=0,\ldots,\ell\,.$

In other words,

$\sum_{s\in{\mathbb{Z}}}\,q_{s}\,s^{r}\,(-1)^{s}\quad=\quad 0,\qquad
r=0,\ldots,\ell\,.$ (18)

Observe that $\sum_{s}q_{s}\,=\,{\boldsymbol{q}}({\boldsymbol{0}})\,=\,1$. For
an arbitrary $h\in{\mathbb{R}}$, we consider the vector of moments
$\tilde{\boldsymbol{h}}=\bigl{(}1,h,\ldots,h^{\ell}\bigr{)}\in{\mathbb{R}}^{\ell+1}$.
Every $\,N\leq\ell+1$ different vector of moments are linearly independent,
since they can be complemented to $\ell+1$ columns of a Vandermonde matrix.
Assume that among the numbers
$\bigl{\\{}s\,=\,(2{\boldsymbol{v}},{\boldsymbol{k}})\,:\
{\boldsymbol{k}}\in{\mathbb{Z}}^{n}\,,c_{{\boldsymbol{k}}}\neq 0\bigr{\\}}$
there are exactly $N$ different ones. Denote them by $s_{1},\ldots,s_{N}$.
Then equation (18) reads
$\sum_{j=1}^{N}(-1)^{s_{j}}\,q_{s_{j}}\,\tilde{\boldsymbol{s}}_{j}\,=\,0$.
Therefore, the number $N$ of vectors $\tilde{\boldsymbol{s}}_{j}$ is at least
$\ell+2$. On the other hand, each nonzero coefficient $c_{{\boldsymbol{k}}}$
generates exactly one number $s=(2{\boldsymbol{v}},{\boldsymbol{k}})$ (which
may coincide for different ${\boldsymbol{k}}$), hence, there are at least
$N\geq\ell+2$ nonzero coefficients $c_{{\boldsymbol{k}}}$.

$\Box$

###### Remark 8.2

The tile B-spline is not the only optimal refinable function. One may classify
all refinement equations with $\nu(\varphi)\geq\ell$ and with $\ell+2$ nonzero
coefficients. We take arbitrary $\ell+2$ numbers $s_{1},\ldots,s_{\ell+2}$,
not of the same parity, and for each $i=1,\ldots,\ell+2$, find an integer
point ${\boldsymbol{k}}_{i}$ such that
$(2{\boldsymbol{v}},{\boldsymbol{k}}_{i})=s_{i}$. This is possible because
${\boldsymbol{v}}\notin{\mathbb{Z}}^{n}$, and hence $2{\boldsymbol{v}}\notin
2{\mathbb{Z}}^{n}$. Then find the solution $q_{s_{1}},\ldots,q_{s_{\ell+2}}$
of the equation (18). We choose the parities of $s_{i}$ so that
$\sum_{i}q_{i}\neq 0$. Normalizing the solution as $\sum_{i}q_{i}=1$ and
taking $c_{{\boldsymbol{k}}_{i}}=2q_{i}$ we obtain the desirable refinement
equation.

##  9 Tile subdivision schemes

We restrict ourselves to subdivision schemes on rectangular grid. From the
results of Section 8 we obtain the following optimal property of schemes
generated by tile B-splines.

###### Theorem 9.1

If a two-digit subdivision scheme preserves the space of algebraic polynomials
of degree $\ell\geq 0$, then it has at least $\ell+2$ coefficients.

Thus, the tile subdivision schemes possess the minimal possible number of
coefficients among all schemes with a given rate of approximation. Moreover,
some of them are super-smooth. The following result was obtained in [51].

Theorem C. Let $\varphi_{\ell}^{(b)}$ be the B-spline generated by the Bear
tile. Then $\varphi_{2}^{(b)}\in C^{2}({\mathbb{R}}^{2})$ and
$\varphi_{3}^{(b)}\in C^{3}({\mathbb{R}}^{2})$.

Note that the cardinal B-splines of orders $2$ and $3$ do not belong to
$C^{2}$ and to $C^{3}$ respectively. In view of Theorem 9.1, this means that
$\varphi_{2}^{(b)}$ has the minimal number (in this case, four) of
coefficients among all $C^{2}$ refinable functions and the same is for
$\varphi_{3}^{(b)}$ among $C^{3}$ functions (five coefficients). The
corresponding tile subdivision schemes involve the minimal number of points of
the grid ($4$ and $5$ respectively) in each iteration, among all $C^{2}$ and
$C^{3}$ schemes.

The coefficients of the mask of the Bear scheme are located on one line:
$c_{{\boldsymbol{k}}},\,\,{\boldsymbol{k}}=(s,0),\,s=0,\ldots,\ell+1$. The
dilation matrix $M_{b}$ given by (16) defines a composition of the
$\sqrt{2}$-expansion and the elliptic rotation by the angle $\beta={\rm
atan}\,\sqrt{7}$. Since $\beta$ is irrational modulo $\pi$, the images of the
integer grid $M^{-k}{\mathbb{Z}}^{2}$ produced with iterations of the
subdivision scheme covers an everywhere dense set of directions. The first
iterations of the $\varphi_{3}^{(b)}$ subdivision scheme ($C^{3}$ scheme with
five coefficients) are shown in figures 5, 6.

Figure 5: First iteration of the $\varphi_{3}^{(b)}$ scheme

Figure 6: Second iteration

Appendix

Proof of Lemma 5.1. Clearly, the operator ${\mathcal{T}}$ respects the
positivity of the function. Therefore, if the space of polynomials
$P_{\ell,s}$ is invariant with respect to ${\mathcal{T}}$, then so is the cone
$K_{\ell,s}$. It is well-known that the operator ${\mathcal{T}}$ respects the
space of polynomials ${\mathcal{P}}_{\Omega}$. The space $P_{\ell,s}$ is also
invariant, provided the sum rules of order $\ell$ are satisfied. Indeed, if a
polynomial ${\boldsymbol{p}}$ has zero of order at least $2(\ell+1)$ at the
point ${\boldsymbol{\xi}}={\boldsymbol{0}}$, then so does the polynomial
${\mathcal{T}}\,{\boldsymbol{p}}\,$: for all digits
${\boldsymbol{d}}^{*}\neq{\boldsymbol{0}}$, the function
$\bigl{|}{\boldsymbol{c}}\,\bigl{(}M^{-T}({\boldsymbol{\xi}}+{\boldsymbol{d}}^{*})\bigr{)}\bigr{|}^{2}$
has zero of order at least $2(\ell+1)$ at the point
${\boldsymbol{\xi}}={\boldsymbol{0}}$ (because of the sum rules), and for
${\boldsymbol{d}}^{*}={\boldsymbol{0}}$ so does the function
${\boldsymbol{p}}\,\bigl{(}M^{-T}({\boldsymbol{\xi}}+{\boldsymbol{d}}^{*})\bigr{)}$
(because of the assumption). Hence, all terms in the sum (3) have zero of
order at least $2(\ell+1)$ at the point ${\boldsymbol{\xi}}={\boldsymbol{0}}$.

It remains to show that if ${\boldsymbol{p}}$ has a root
${\boldsymbol{z}}\in\pi\,\bigl{\\{}\mbox{\em
nill}\,(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$ of order at least
$2(\ell+1)$, then so has the polynomial ${\mathcal{T}}{\boldsymbol{p}}$. To
this end we consider arbitrary vectors ${\boldsymbol{h}}_{i}\in
J_{s},\,i=0,\ldots,\ell$, and denote by
$g=\varphi_{\bar{\boldsymbol{h}}}\,=\,\frac{\partial}{\partial{\boldsymbol{h}}_{0}\cdots\partial{\boldsymbol{h}}_{\ell}}\,\varphi$
the mixed derivative of order $\ell+1$ of the refinable function $\varphi$
along the directions ${\boldsymbol{h}}_{i}$. The function $\Phi_{g}$ is a
polynomial from ${\mathcal{P}}_{\Omega}$ which has a root of order at least
$2(\ell+1)$ at the point ${\boldsymbol{z}}$. Indeed,
$\widehat{g}({\boldsymbol{\xi}})=\widehat{\varphi_{{\boldsymbol{h}}}}({\boldsymbol{\xi}})\,=\,2\pi
i({\boldsymbol{h}},{\boldsymbol{\xi}})\widehat{\varphi}({\boldsymbol{\xi}})$.
If an integer shift ${\boldsymbol{z}}+{\boldsymbol{k}}$ belongs to $\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})$, then this point is a root of
$\widehat{\varphi}$ of order at least $\ell+1$; if
${\boldsymbol{z}}+{\boldsymbol{k}}\in J_{s}^{\perp}$, then
$({\boldsymbol{h}}_{i},{\boldsymbol{z}}+{\boldsymbol{k}})=0$ for all $i$, and
we again have a zero of order at least $\ell+1$ of the function $\widehat{g}$
at the point ${\boldsymbol{z}}+{\boldsymbol{k}}$. Hence, $\Phi_{g}$ have zeros
of order at least $\ell+1$ at all the points
${\boldsymbol{z}}+{\boldsymbol{k}},\ {\boldsymbol{k}}\in{\mathbb{Z}}^{n}$.
This is true for every $z\,\in\,\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,\cup\,J_{s}^{\perp}$, therefore, $\Phi_{g}$
vanishes at the set $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$. On the other
hand, ${\mathcal{T}}(\Phi_{g})\,=\,\Phi_{{\mathcal{F}}g}$, where the function
${\mathcal{F}}g\,=\,{\mathcal{F}}(\varphi_{\bar{\boldsymbol{h}}})\,=\,\varphi_{M\bar{\boldsymbol{h}}}$.
Therefore,
${\mathcal{T}}(\Phi_{g})\,=\,\Phi_{\varphi_{M\bar{\boldsymbol{h}}}}$ with
$M\bar{\boldsymbol{h}}\in J_{s}^{\,\ell}$, because $J_{s}$ is an invariant
subspace of $M$. Hence, the polynomial ${\mathcal{T}}(\Phi_{g})$ also vanishes
on the set $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$. It remains to
note that zeros of the polynomial ${\mathcal{T}}{\boldsymbol{p}}$ depend only
on the location and orders of zeros of ${\boldsymbol{p}}$. Hence, if a
polynomial vanishes on $\pi\,\bigl{\\{}\mbox{\em
nill}\,_{\ell}(\widehat{\varphi})\,,\,J_{s}^{\perp}\bigr{\\}}$, then so does
${\mathcal{T}}{\boldsymbol{p}}$, which completes the proof.

$\Box$

Algorithms and implementation details

To write the matrix of ${\mathcal{T}}$ we need to find the set
$\Omega=\bigl{(}Y-Y\bigr{)}\cap{\mathbb{Z}}^{n}$, which can be obtained by a
special iterative algorithms, similar to ones presented in [10, 11]. We give a
brief description below.

Using the definition of $Y$ (subsection 5.1), we obtain

$Y-Y\quad=\quad\left\\{\
\sum_{j=1}^{\infty}M^{-j}\,({\boldsymbol{s}}_{j}-{\boldsymbol{t}}_{j})\
:\quad{\boldsymbol{s}}_{j},{\boldsymbol{t}}_{j}\in Q\,,\ j\in{\mathbb{N}}\
\right\\}\,.$

Denote $Q^{\prime}=Q-Q$ and rewrite:

$Y-Y\quad=\quad\left\\{\ \sum_{j=1}^{\infty}M^{-j}\,{\boldsymbol{s}}_{j}\
:\quad{\boldsymbol{s}}_{j}\in Q^{\prime}\,,\ j\in{\mathbb{N}}\ \right\\}\,.$

Define the map ${\mathcal{J}}$ on the set of compact subsets
$X\subset{\mathbb{R}}^{n}$ as follows:
${\mathcal{J}}\,X\,=\,M^{-1}(X+Q^{\prime})$. A set $X$ is invariant with
respect to ${\mathcal{J}}$, if ${\mathcal{J}}\,X\,\subset\,X$. Similarly we
define a map ${\mathcal{J}}_{0}$ on the set of finite subsets of integers
$X\subset{\mathbb{Z}}^{n}$:
${\mathcal{J}}_{0}\,X\,=\,{\mathcal{J}}X\cap{\mathbb{Z}}^{n}$. The invariance
with respect to ${\mathcal{J}}_{0}$ is defined in the same way.

###### Proposition 9.2

The set $\Omega$ is invariant with respect to $J_{0}$.

Proof. Let ${\boldsymbol{z}}$ be an arbitrary element of the set
${\mathcal{J}}_{0}\,\Omega$, i.e.,
${\boldsymbol{z}}=M^{-1}({\boldsymbol{s}}+{\boldsymbol{q}}^{\prime})$,
${\boldsymbol{z}}\in{\mathbb{Z}}^{n}$ for some
${\boldsymbol{s}}\in\Omega,{\boldsymbol{q}}^{\prime}\in Q^{\prime}$. Since
${\boldsymbol{s}}\in\Omega$, we have
${\boldsymbol{s}}=\sum_{j=1}^{\infty}M^{-j}\,{\boldsymbol{s}}_{j}$,
${\boldsymbol{s}}_{j}\in Q^{\prime}$. Therefore
${\boldsymbol{z}}=M^{-1}({\boldsymbol{s}}^{\prime}+{\boldsymbol{q}}^{\prime})=\sum_{j=1}^{\infty}M^{-j}\,{\boldsymbol{s}}_{j}^{\prime}$,
where ${\boldsymbol{s}}_{1}^{\prime}={\boldsymbol{q}}^{\prime}$,
${\boldsymbol{s}}_{j}^{\prime}={\boldsymbol{s}}_{j-1}$ for $j\geq 2$. Thus,
${\boldsymbol{z}}\in Y-Y$. Since ${\boldsymbol{z}}\in{\mathbb{Z}}^{n}$, it
follows that ${\boldsymbol{z}}\in\Omega$, which completes the proof.

$\Box$

To find the set $\Omega$, we start with a subset
$\Omega_{0}\subset{\mathbb{Z}}^{n}$, which will be specified later, then
define $\Omega_{i+1}\,=\,\Omega_{i}\,\cup\,{\mathcal{J}}_{0}\Omega_{i}$ for
$i\geq 0$. Thus, we obtain a non-increasing (by inclusion) sequence
$\\{\Omega_{i}\\}_{i\geq 0}$. Since $M$ is an expanding matrix, it follows
that there exist $\varepsilon>0$ and an ellipsoid $E\subset{\mathbb{R}}^{n}$
(Lyapunov ellipsoid) such that $M^{-1}E\subset(1-\varepsilon)E$. Hence, for
sufficiently large $\lambda$, we have $M^{-1}(\lambda
E+Q^{\prime})\,\subset\lambda E$. This implies that the sequence
$\\{\Omega_{i}\\}_{i\geq 0}$ is bounded above: all sets $\Omega_{i}$ are
inside $\lambda E$, where $\lambda$ is large enough. Therefore, for some $k$,
we have $\Omega_{k+1}=\Omega_{k}$. It remains to put $\Omega=\Omega_{k}$.

The set $\Omega_{0}$ can be defined as $H+\Delta$, where $\Delta$ is the set
of integer vectors ${\boldsymbol{x}}$ such that $\sum_{i=1}^{n}|x_{i}|\leq 1$,
and $H$ is an arbitrary compact set such that ${\mathcal{J}}H\,\subset\,H$
[43]. For example, one may take the aforementioned Lyapunov ellipsoid $E$,
which can be found by solving the corresponding semidefinite programming
problem.

Acknowledgements. The authors are grateful to S.Tikhonov and I.Shkredov for
useful discussions.

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