Positive Definite Functions on Complex Spheres and their Walks through
Dimensions

Positive Definite Functions on Complex Spheres
and their Walks through Dimensions

Eugenio MASSA †, Ana Paula PERON † and Emilio PORCU ‡§

E. Massa, A.P. Peron and E. Porcu

† Departamento de Matemática, ICMC-USP - São Carlos,
† Caixa Postal 668, 13560-970 São Carlos SP, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ School of Mathematics and Statistics, Chair of Spatial Analytics Methods,
‡ University of Newcastle, UK<EMAIL_ADDRESS>
§ Department of Mathematics, Universidad Técnica Federico Santa Maria,
§ Avenida España 1680, Valparaíso, 230123, Chile

Received April 06, 2017, in final form October 30, 2017; Published online
November 08, 2017

We provide walks through dimensions for isotropic positive definite functions
defined over complex spheres. We show that the analogues of Montée and
Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory
221 (2017), 22–37] on the basis of the original Matheron operator [Les
variables régionalisées et leur estimation, Masson, Paris, 1965], allow for
similar walks through dimensions. We show that the Montée operators also
preserve, up to a constant, strict positive definiteness. For the Descente
operators, we show that strict positive definiteness is preserved under some
additional conditions, but we provide counterexamples showing that this is not
true in general. We also provide a list of parametric families of (strictly)
positive definite functions over complex spheres, which are important for
several applications.

Descente; disk polynomials; Montée; positive definite functions

42A82; 42C10; 42C05; 30E10; 62M30

##  1 Introduction and main results

Positive definite functions have a long history which can be traced back to
papers by Carathéodory, Herglotz, Bernstein and Matthias, culminating in
Bochner’s theorem from 1932–1933. See Berg [6] for details. In the last twenty
years several results related to this topic were obtained in fields as diverse
as mathematical analysis, numerical analysis, potential theory, probability
theory and geostatistics: we refer the reader to the surveys in Schaback [35,
36], Berg [6] and Fasshauer [14] for a complete list of references in this
direction.

Positive definite radial functions have been known since the two seminal
papers by Schoenberg [39, 40]. The former is devoted to radially symmetric
functions depending on the Euclidean distance, and the latter to isotropic
functions on unit spheres ${\mathbb{S}}^{d}$ of $\mathbb{R}^{d+1}$. Literature
on radially symmetric functions on Euclidean spaces has been especially
fervent. In his essay devoted to the clavier spherique, Matheron [24] proposed
operators called Montée and Descente that preserve the property of positive
definiteness but changing the dimension of the space initially considered.
Such a property has been called walk through dimensions. It is worth noting
that the walk through dimensions is achieved at the expense of modifying the
differentiability at the origin of a given candidate function. Wendland [45]
used the Montée operator with a class of compactly supported radial basis
functions, termed Wendland’s functions after his works. Schaback [37] covered
the missing cases of walks through dimensions. Porcu et al. [30] used a
fractional version of the Montée operator to obtain generalized versions of
Wendland’s functions. For a reference on walks through dimensions in the
geostatistical setting, the reader is referred to Gneiting [16] and to the
more recent work of Porcu and Zastavnyi [29].

Positive definite functions as well as strictly positive definite functions in
several contexts have been deeply studied by the mathematical analysis
literature, and the reader is referred to the works by Menegatto et al. (see
Chen et al. [11], Menegatto and Peron [26], Guella et al. [19], and references
therein). The use of positive definite functions on real spheres for
geostatisticians has arrived recently, thanks to the survey by Gneiting [17]
and the recent developments by Berg and Porcu [7] and Porcu et al. [28]. In
particular, Berg and Porcu [7] characterized the class of the positive
definite functions on the product of ${\mathbb{S}}^{d}$ with a locally compact
group, extending the Schoenberg’s class $\Psi_{d}$ of the positive definite
functions on ${\mathbb{S}}^{d}$ (Schoenberg [40]).

A continuous function $f\colon[-1,1]\to\mathbb{R}$ belongs to the class
$\Psi_{d}$ when the kernel

$\displaystyle
K\colon{\mathbb{S}}^{d}\times{\mathbb{S}}^{d}\to\mathbb{R}\colon
K(\xi,\eta)=f(\langle\xi,\eta\rangle)$

is positive definite. Schoenberg [40] proved that $f\in\Psi_{d}$ if, and only
if,

$\displaystyle f(x)=\sum_{k\geq 0}a_{k}^{d}c_{k}(d,x),\qquad\sum_{k\geq
0}a_{k}^{d}<\infty,\qquad a_{k}^{d}\geq 0,\quad\forall\,k\geq 0,$ (1.1)

where $c_{k}(d,\cdot)$ are the normalized Gegenbauer polynomials associated to
the index $d$ (see Szegő [44, p. 80]). The coefficients in the above series
are called $d$-Schoenberg coefficients. On the other hand, the subclass
$\Psi_{d}^{+}$ of $\Psi_{d}$ of the strict positive definite functions on
${\mathbb{S}}^{d}$, $d\geq 2$, was characterized by Chen et al. [11]:
$f\in\Psi_{d}^{+}$ if, and only if, the set $\\{k\colon a_{k}^{d}>0\\}$
contains infinitely many odd and infinitely many even integers.

The class $\Psi_{d}$ has received special interest in the last twenty years,
while walks through dimensions for positive definite functions on real spheres
have been studied in the recent tour de force by Beatson and zu Castell [3,
4]. In particular, Beatson and zu Castell [4] define the Montée operator

$\displaystyle(If)(x)=\int_{-1}^{x}f(u){\rm d}u,\qquad x\in[-1,1],$

for $f$ integrable in $[-1,1]$, and the Descente operator

$\displaystyle(Df)(x)=\frac{{\rm d}}{{\rm d}x}f(x),\qquad x\in[-1,1],$

for $f$ absolutely continuous in $[-1,1]$. They prove that, for $d\geq 2$:

  1. (i)

if $f\in\Psi_{d+2}$, then there exists a constant $c$ such that
$c+If\in\Psi_{d}$;

  2. (ii)

if $f\in\Psi_{d+2}^{+}$, then there exists a constant $c$ such that
$c+If\in\Psi_{d}^{+}$;

  3. (iii)

if $f\in\Psi_{d+2}$, $f\geq 0$ and all $(d+2)$-Schoenberg coefficients are
positive, then $If\in\Psi_{d}$ and all its $d$-Schoenberg coefficients are
positive;

  4. (iv)

if $f\in\Psi_{d}$ and $Df$ is continuous, then $Df\in\Psi_{d+2}$;

  5. (v)

if $f\in\Psi_{d}^{+}$ and $Df$ is continuous, then $Df\in\Psi_{d+2}^{+}$.

Observe that the property of (strict) positive definiteness of $f$ is
preserved by the operators Montée $I$ and Descente $D$.

In this paper, inspired by the work of Beatson and zu Castell [4], we study
positive definite functions on complex unit spheres $\Omega_{2q}$ of
$\mathbb{C}^{q}$. In particular, we provide walks through dimensions over
complex spheres.

Below, we state our main results and we refer to Section 2 for the necessary
background.

We denote the class of positive definite functions on $\Omega_{2q}$ by
$\Psi(\Omega_{2q})$. A characterization of such functions was proposed in
Menegatto and Peron [26]: let ${\mathbb{D}}:=\\{z\in\mathbb{C}\colon|z|\leq
1\\}\subset\mathbb{C}$, when a continuous function
$f\colon{\mathbb{D}}\to\mathbb{C}$ belongs to $\Psi(\Omega_{2q})$, an
expansion similar to (1.1) exists, namely

$\displaystyle f(z)=\sum_{m,n\geq 0}a_{m,n}^{q-2}R_{m,n}^{q-2}(z),\qquad
z\in{\mathbb{D}},$

(see equation (2.3) and Theorem 2.1). We will call the coefficients
$a_{m,n}^{q-2}$ as $(2q)$-complex Schoenberg coefficients.

In order to make the statements clear, it is convenient to introduce the
Descente and Montée operators in the complex context.

Given $f\colon{\mathbb{D}}\to\mathbb{C}$, we say that $f$ is differentiable
if, writing $z=x+iy\in{\mathbb{D}}$, $f$ is differentiable as a function of
$x$ and $y$. Then, we denote by ${\cal D}_{x}f$ and ${\cal D}_{y}f$ the
partial derivatives with respect to $x$ and $y$, respectively, and we define
the Descente operators through the following Wirtinger derivatives:

$\displaystyle{\cal D}_{z}f=\frac{1}{2}({\cal D}_{x}f-i{\cal
D}_{y}f),\qquad{\cal D}_{\overline{z}}f=\frac{1}{2}({\cal D}_{x}f+i{\cal
D}_{y}f).$ (1.2)

We observe that $f$ might not be complex differentiable, actually it is so
only when ${\cal D}_{\overline{z}}f=0$, and in this case ${\cal
D}_{z}f=f^{\prime}$, the complex derivative of $f$.

If $f$ admits a $z$-primitive $F$ and a $\overline{z}$-primitive $G$ in
${\mathbb{D}}$, that is, ${\cal D}_{z}F={\cal D}_{\overline{z}}G=f$, then we
can define the Montée operators ${\cal I}$ and ${\cal\overline{I}}$ by

$\displaystyle{\cal
I}(f)(z):=F(z)-F(0)\qquad\text{and}\qquad{\cal\overline{I}}(f)(z):=G(z)-G(0),\qquad
z\in{\mathbb{D}}.$

By definition,

$\displaystyle{\cal D}_{z}({\cal I}f)=f\qquad\text{and}\qquad{\cal
D}_{\overline{z}}(\overline{{\cal I}}f)=f.$ (1.3)

Moreover,

$\displaystyle{\cal I}({\cal
D}_{z}(f))(z)=f(z)-f(0)\qquad\text{and}\qquad\overline{{\cal I}}({\cal
D}_{\overline{z}}(f))(z)=f(z)-f(0),\qquad z\in{\mathbb{D}}.$

Our main results are related with walks through dimensions for Descente and
Montée operators over complex spheres:

######  Theorem 1.1.

Let $f\colon{\mathbb{D}}\to\mathbb{C}$ be continuously differentiable.

  1. $(i)$

If $f$ belongs to the class $\Psi(\Omega_{2q})$, then ${\cal D}_{z}f$, ${\cal
D}_{\overline{z}}f$ and ${\cal D}_{x}f$ belong to the class
$\Psi(\Omega_{2q+2})$.

  2. $(ii)$

If $f$ belongs to the class $\Psi(\Omega_{2q})$ and has all positive
$(2q)$-complex Schoenberg coefficients, then ${\cal D}_{z}f$, ${\cal
D}_{\overline{z}}f$ and ${\cal D}_{x}f$ belong to the class
$\Psi^{+}(\Omega_{2q+2})$.

######  Theorem 1.2.

Let $f\colon{\mathbb{D}}\to\mathbb{C}$ be a continuous function admitting a
$z$-primitive and a $\overline{z}$-primitive in ${\mathbb{D}}$.

  1. $(i)$

If $f$ belongs to the class $\Psi(\Omega_{2q+2})$, then there exist real
constants $c$ and $C$ such that $c+{\cal I}f$ and $C+{\overline{\cal I}}f$
belong to the class $\Psi(\Omega_{2q})$.

  2. $(ii)$

If $f$ belongs to the class $\Psi^{+}(\Omega_{2q+2})$, then there exist real
constants $c$ and $C$ such that $c+{\cal I}f$ and $C+{\overline{\cal I}}f$
belong to the class $\Psi^{+}(\Omega_{2q})$.

Observe that in Theorem 1.1(ii) we assumed the additional condition that all
$(2q)$-complex Schoenberg coefficients are positive. This condition can be
weakened (see Remark 1.4 below), but not completely removed.

In fact, the following counterexamples show that the Descente operators over
complex spheres do not preserve, in general, strict positive definiteness, in
contrast to the real case of Beatson and zu Castell.

######  Counterexample 1.3.

Let $q\geq 2$ be an integer.

  1. $(i)$

If $f(z)=\sum\limits_{m=0}^{\infty}a_{m,0}^{q-2}R_{m,0}^{q-2}(z)$, where
$\sum\limits_{m=0}^{\infty}a_{m,0}^{q-2}<\infty$ and $a_{m,0}^{q-2}>0$ for all
$m$, then $f\in\Psi^{+}(\Omega_{2q})$ and ${\cal D}_{z}f,{\cal
D}_{x}f\in\Psi^{+}(\Omega_{2q+2})$ but ${\cal
D}_{\overline{z}}f\not\in\Psi^{+}(\Omega_{2q+2})$.

  2. $(ii)$

If $f(z)=\sum\limits_{n=0}^{\infty}a_{0,n}^{q-2}R_{0,n}^{q-2}(z)$, where
$\sum\limits_{n=0}^{\infty}a_{0,n}^{q-2}<\infty$ and $a_{0,n}^{q-2}>0$ for all
$n$, then $f\in\Psi^{+}(\Omega_{2q})$ and ${\cal D}_{\overline{z}}f,{\cal
D}_{x}f\in\Psi^{+}(\Omega_{2q+2})$ but ${\cal
D}_{z}f\not\in\Psi^{+}(\Omega_{2q+2})$.

  3. $(iii)$

If
$f(z)=\sum\limits_{n=0}^{\infty}a_{0,n}^{q-2}R_{0,n}^{q-2}(z)+\sum\limits_{m=0}^{\infty}a_{m,0}^{q-2}R_{m,0}^{q-2}(z)$,
where $a_{0,n}^{q-2},a_{m,0}^{q-2}\geq 0$ for all $m$, $n$, and

$\displaystyle a_{0,n}^{q-2}>0\Longleftrightarrow n\in 5\mathbb{Z}_{+}+4,$
$\displaystyle a_{m,0}^{q-2}>0\Longleftrightarrow
m\in(5\mathbb{Z}_{+}\setminus\\{0\\})\cup(5\mathbb{Z}_{+}+2)\cup(5\mathbb{Z}_{+}+3)\cup(5\mathbb{Z}_{+}+4),$

then $f\in\Psi^{+}(\Omega_{2q})$ but ${\cal D}_{z}f,{\cal
D}_{\overline{z}}f,\mathcal{D}_{x}f\notin\Psi^{+}(\Omega_{2q+2})$.

######  Remark 1.4.

In the real case, the condition that all $d$-Schoenberg coefficients are
positive is satisfied by most of the functions in the class $\Psi_{d}^{+}$
which appear in applications such as in statistics and geostatistics.

In the complex case, among the examples that we provide in Section 2.1, only
the exponential function satisfies this condition. On the other hand, the
Aktaş–Taşdelen–Yavuz, Horn and Lauricella families, satisfy the following
simple weaker condition, which is also sufficient to obtain the conclusion of
Theorem 1.1(ii):

  * •

if $a_{m,n}^{q-2}$ are the $(2q)$-complex Schoenberg coefficients of $f$, then
for some $c,d\in\mathbb{N}$, the set

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-2}>0,\,m,n\geq c\big{\\}}$

contains $(d+\mathbb{Z}_{+})$ or $(-d-\mathbb{Z}_{+})$.

In fact, the weakest possible condition to be used in Theorem 1.1(ii) follows
from Guella and Menegatto [18] and reads as follows:

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-2}>0,\,m,n\geq
1\big{\\}}\cap(N\mathbb{Z}+j)\neq\varnothing,$ (1.4)

for every $N\geq 1$, $j=0,1,\ldots,N-1$. We will prove Theorem 1.1 with this
last condition, since the previous ones are stronger.

This paper is organized as follows: in Section 2, we provide the necessary
background about positive definite functions on complex spheres and we give a
list of parametric families of these functions, which are of interest for both
numerical analysis and geostatistical communities. Finally, in Section 3, we
obtain all necessary technical lemmas, we give the proofs of Theorems 1.1 and
1.2, and we show the Counterexample 1.3.

##  2 The classes $\boldsymbol{\Psi(\Omega_{2q})}$ and
$\boldsymbol{\Psi^{+}(\Omega_{2q})}$: a brief survey

This section is largely expository and presents some basic facts and
background needed for a self contained exposition.

For $q$ being a positive integer, we denote by $\Omega_{2q}$ the unit sphere
of $\mathbb{C}^{q}$ and by $B_{2q}:=\\{z\in\mathbb{C}^{q}\colon|z|\leq 1\\}$
the closed disk in $\mathbb{C}^{q}$. Also, we define the Pochhammer symbol
$(a)_{n}:=a(a+1)\cdots(a+n-1)$, with $(a)_{0}:=1$.

Let $A$ be a nonempty set. A continuous kernel $K\colon
A^{2}\rightarrow\mathbb{C}$ is positive definite if and only if

$\displaystyle\sum_{\mu,\nu=1}^{l}c_{\mu}\overline{c_{\nu}}K(\xi_{\mu},\xi_{\nu})\geq
0,$ (2.1)

for all $l\in\mathbb{Z}_{+}:=\\{0,1,2,\ldots\\}$,
$\\{\xi_{1},\xi_{2},\ldots,\xi_{l}\\}\subset A$ and
$\\{c_{1},c_{2},\ldots,c_{l}\\}\subset\mathbb{C}$. If the inequality in (2.1)
is strict when at least one $c_{\mu}$ is nonzero, then $K$ is called strictly
positive definite. For $q$ a strictly positive integer, we define
$A_{q}:=\Omega_{2}$ when $q=1$ and $A_{q}:={\mathbb{D}}$ for $q>1$. Throughout
we shall work with the class $\Psi(\Omega_{2q})$ of continuous functions
$f\colon A_{q}\to\mathbb{C}$ such that the kernel
$K\colon\Omega_{2q}\times\Omega_{2q}\to\mathbb{C}$ defined as

$\displaystyle
K(\xi,\eta)=f(\langle\xi,\eta\rangle),\qquad(\xi,\eta)\in\Omega_{2q}\times\Omega_{2q},$
(2.2)

where the symbol $\langle\cdot,\cdot\rangle$ denotes the usual inner product
in $\mathbb{C}^{q}$, is positive definite.

Observe that an immediate consequence of the definition is that $f$ satisfies
$f(\overline{z})=\overline{f(z)}$. We shall use the notation
$\Psi^{+}(\Omega_{2q})$ if the kernel $K$ associated to $f$ through (2.2) is
strictly positive definite. Positive definite kernels satisfying the identity
above are called isotropic. The class $\Psi(\Omega_{2q})$ is parenthetical to
the class $\Psi_{d}$ introduced by Schoenberg [40], and we refer the reader to
the recent review in Gneiting [17] for a thorough description of the
properties of this class. Further, the class $\Psi_{d}$ represents the
building block for extension to product spaces, and the reader is referred to
Berg and Porcu [7] as well as to Guella et al. [19] for recent efforts in this
direction. The classes $\Psi(\Omega_{2q})$ are nested, with the following
inclusion relation being strict:

$\displaystyle\Psi(\Omega_{4})\supset\Psi(\Omega_{6})\supset\cdots\supset\Psi(\Omega_{\infty}),$

where $\Omega_{\infty}$ is the unit sphere in the Hilbert space
$\ell_{2}(\mathbb{C})$. Analogous relations apply to $\Psi^{+}(\Omega_{2q})$.

Observe that the class $\Psi(\Omega_{2})$ is a different class and it can not
be added to the inclusions above (see Menegatto and Peron [26]). For this
reason, in this work we always consider $q\geq 2$. Actually the main purpose
here is to study the walks through dimensions considering functions in the
classes $\Psi(\Omega_{2q})$.

Characterization theorems for the classes $\Psi(\Omega_{2q})$ are available in
recent literature, and some ingredients are needed for a detailed exposition.
We refer to Boyd and Raychowdhury [10], Dreseler and Hrach [13], and
Koornwinder [22, 23] for more information concerning this necessary material.

The disc polynomial $R_{m,n}^{\alpha}$ of degree $m+n$ in $x$ and $y$
associated to a real number $\alpha>-1$ was introduced by Zernike [47] and
Zernike and Brinkman [48], see also Koornwinder [22], as the polynomial given
by

$\displaystyle
R_{m,n}^{\alpha}(z):=r^{|m-n|}e^{i(m-n)\theta}R_{\min\\{m,n\\}}^{(\alpha,|m-n|)}\big{(}2r^{2}-1\big{)},\qquad
z=re^{i\theta}=x+iy\in{\mathbb{D}},$ (2.3)

where $R_{k}^{(\alpha,\beta)}$ is the usual Jacobi polynomial of degree $k$
associated to the numbers $\alpha,\beta>-1$ and normalized by
$R_{k}^{(\alpha,\beta)}(1)=1$ (see Szegő [44, p. 58]). Note that the function
$R_{m,n}^{\alpha}$ is a polynomial of degrees $m$ and $n$ with respect to the
arguments $z$ and $\overline{z}$, respectively. Moreover it satisfies
$R_{m,n}^{\alpha}(\overline{z})=\overline{R_{m,n}^{\alpha}(z)}$.

Let ${\rm d}\nu_{\alpha}$ be the positive measure having total mass
identically equal to one on ${\mathbb{D}}$, and given by

$\displaystyle{\rm
d}\nu_{\alpha}(z)=\frac{\alpha+1}{\pi}\big{(}1-x^{2}-y^{2}\big{)}^{\alpha}{\rm
d}x{\rm d}y,\qquad z=x+iy.$ (2.4)

Due to the orthogonality relations for Jacobi polynomials, the set
$\\{R_{m,n}^{\alpha}\colon 0\leq m,n<\infty\\}$ forms a complete orthogonal
system in $L^{2}({\mathbb{D}},{\rm d}\nu_{\alpha})$ with

$\displaystyle\int_{{\mathbb{D}}}R_{m,n}^{\alpha}(z)\overline{R_{k,l}^{\alpha}(z)}{\rm
d}\nu_{\alpha}(z)=\frac{1}{h_{m,n}^{\alpha}}\delta_{m,k}\delta_{n,l},$ (2.5)

where

$\displaystyle
h_{m,n}^{\alpha}=\frac{m+n+\alpha+1}{\alpha+1}\left(\begin{matrix}\alpha+m\\\
\alpha\end{matrix}\right)\left(\begin{matrix}\alpha+n\\\
\alpha\end{matrix}\right),$ (2.6)

and $\delta_{n,l}$ denotes the Kronecker delta. Thus, a function $f\in
L^{1}({\mathbb{D}},\nu_{\alpha})$, $\alpha\geq 0$, has an expansion in terms
of disc polynomials $R_{m,n}^{\alpha}$ defined through

$\displaystyle f(z)\sim\sum_{m,n\geq 0}a_{m,n}^{\alpha}R_{m,n}^{\alpha}(z),$
(2.7)

where

$\displaystyle
a_{m,n}^{\alpha}=h_{m,n}^{\alpha}\int_{{\mathbb{D}}}f(z)\overline{R_{m,n}^{\alpha}(z)}{\rm
d}\nu_{a}(z).$ (2.8)

The Poisson–Szegő kernel will be a fundamental tool for the proof of Theorem
2.1(1) below: the characterization of the class $\Psi(\Omega_{2q})$. We give
here a brief presentation of it, since this kernel will also be used ahead.
The Poisson–Szegő kernel is defined by

$\displaystyle{\cal
P}_{q}(r\xi,\eta):=\frac{1}{\sigma_{2q}}\frac{(1-|r\xi|^{2})^{q}}{|1-\langle
r\xi,\eta\rangle|^{2q}},\qquad r\in[0,1),\quad\xi,\eta\in\Omega_{2q},$ (2.9)

where $\sigma_{2q}$ is the total surface of $\Omega_{2q}$. Folland [15] proved
that it has an expansion in terms of disc polynomials as

$\displaystyle{\cal P}_{q}(r\xi,\eta)=\sum_{m,n\geq
0}\frac{h_{m,n}^{q-2}}{\sigma_{2q}}S_{m,n}^{q}(r)R_{m,n}^{q-2}(\langle\xi,\eta\rangle),\qquad\xi,\eta\in\Omega_{2q},\quad
r\in[0,1),$ (2.10)

where $S_{m,n}^{q}(r)\geq 0$, $\lim\limits_{r\to 1^{-}}S_{m,n}^{q}(r)=1$ and
the series converges absolutely and uniformly for $\xi,\eta\in\Omega_{2q}$ and
$0\leq r\leq R$, for each $R<1$.

The Poisson–Szegő kernel also appears in the solution of the following
Dirichlet problem for the Laplace–Beltrami operator $\Delta_{2q}$ (see Stein
[43]): given a continuous function $h\colon\Omega_{2q}\rightarrow\mathbb{C}$,
there exists a continuous function $u\colon B_{2q}\rightarrow\mathbb{C}$ such
that $\Delta_{2q}u=0$ and $u|_{\Omega_{2q}}=h$. The solution $u$ can be
computed through

$\displaystyle u(z)=\int_{\Omega_{2q}}{\cal P}_{q}(z,\rho)h(\rho){\rm
d}\omega_{2q}(\rho),\qquad z\in B_{2q},$ (2.11)

where ${\rm d}\omega_{2q}$ denotes the rotation-invariant surface element on
$\Omega_{2q}$.

In fact, using this, if $\alpha=q-2\geq 0$ is an integer and $f$ is a
continuous function on ${\mathbb{D}}$, the coefficients in the series in
(2.7), can be written as (see Menegatto and Peron [26]):

$\displaystyle
a_{m,n}^{q-2}=\frac{h_{m,n}^{q-2}}{\sigma_{2q}}\int_{\Omega_{2q}}f(\langle\rho,e_{1}\rangle)R_{m,n}^{q-2}(\langle
e_{1},\rho\rangle){\rm d}\omega_{2q}(\rho),$ (2.12)

where $e_{1}=(1,0,\ldots,0)\in\Omega_{2q}$.

We give now the representations for the elements of the classes
$\Psi(\Omega_{2q})$ and $\Psi^{+}(\Omega_{2q})$ that were proved by Menegatto
and Peron [25, 26] and Guella and Menegatto [18]:

######  Theorem 2.1.

Let $f\colon{\mathbb{D}}\to\mathbb{C}$ be a continuous function. The following
assertions are true:

  1. $(1)$

$f\in\Psi(\Omega_{2q})$ if, and only if,

$\displaystyle f(z)=\sum_{m,n\geq 0}a_{m,n}^{q-2}R_{m,n}^{q-2}(z),\qquad
z\in{\mathbb{D}},$ (2.13)

where $\sum\limits_{m,n\geq 0}a_{m,n}^{q-2}<\infty$ and $a_{m,n}^{q-2}\geq 0$
for all $(m,n)$;

  2. $(2)$

$f\in\Psi^{+}(\Omega_{2q})$ if, and only if, $f\in\Psi(\Omega_{2q})$ and

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-2}>0,\,m,n\geq
0\big{\\}}\cap(N\mathbb{Z}+j)\neq\varnothing,$ (2.14)

for every $N\geq 1$, $j=0,1,\ldots,N-1$.

Note that the index $\alpha=q-2$ of the disc polynomials is related to the
sphere $\Omega_{2q}$ and consequently $\alpha+1=q-1$ is related to
$\Omega_{2q+2}$.

The coefficients $a_{m,n}^{q-2}$ are the analogue of the $d$-Schoenberg
coefficients $a_{k}^{d}$ as in Daley and Porcu [12] and Ziegel [49], referring
to the expansion of the members of the Schoenberg class $\Psi_{d}$. In
analogy, we will call $a_{m,n}^{q-2}$ as $(2q)$-complex Schoenberg
coefficients.

###  2.1 Families within the classes $\boldsymbol{\Psi(\Omega_{2q})}$ and
$\boldsymbol{\Psi^{+}(\Omega_{2q})}$

It is well known that there exist many examples of functions in the class
$\Psi_{d}$, some of them widely used in applications (see for example Gneiting
[17] and Porcu et al. [28]).

In the literature it is also possible to find examples of functions that
satisfy the conditions in Theorem 2.1, or those in Remark 1.4, and therefore
they belong to the classes $\Psi(\Omega_{2q})$ and $\Psi^{+}(\Omega_{2q})$.
Some of them, as well as their use in applications, appeared recently,
probably originated by the work of Wünsche [46], that deals with disc
polynomials: a fundamental tool for studying the functions in these classes.
We give below a collection of such functions.

1\. Disk Polynomials and related families. The product kernel (Boyd and
Raychowdhury [10]),

$\displaystyle
f_{m,n}(z)=z^{m}\overline{z}^{n}=\sum_{j=0}^{\min\\{m,n\\}}c_{q,m,n}^{j}R_{m-j,n-j}^{q-2}(z),\qquad
c_{q,m,n}^{j}\geq 0,\quad z\in{\mathbb{D}},$

is an element of the class $\Psi(\Omega_{2q})$, for each $m,n\geq 0$.

2\. Poisson–Szegő kernel and related families. An application of (2.9) and
(2.10) shows that

$\displaystyle
f_{r}(z):=\frac{1}{\sigma_{2q}}\frac{(1-r^{2})^{q}}{|1-rz|^{2q}}=\sum_{m,n\geq
0}\frac{h_{m,n}^{q-2}}{\sigma_{2q}}S_{m,n}^{q}(r)R_{m,n}^{q-2}(z),\quad\
z\in{\mathbb{D}},$

and hence it is a member of the class $\Psi(\Omega_{2q})$, for each
$r\in[0,1)$.

3\. Exponential function. The function (Menegatto et al. [27])

$\displaystyle
e^{z+\overline{z}}=\sum_{m+n=0}^{\infty}\frac{(m+1)_{q-2}(n+1)_{q-2}}{(q-2)!}\left(\sum_{j=0}^{\infty}\frac{1}{j!(m+n+q-1)_{j}}\right)R_{m,n}^{q-2}(z),\qquad
z\in{\mathbb{D}},$

belongs to the class $\Psi^{+}(\Omega_{2q})$.

4\. Aktaş, Taşdelen and Yavuz family. The function (Aktaş et al. [2])

$\displaystyle
f_{t}(z):=\frac{1}{R}\left(\frac{2}{1-t+R}\right)^{q-2}e^{(2tz)/(1+t+R)}=\sum_{m,n\geq
0}(q-1)_{n}\frac{t^{m+n}}{m!n!}R_{m+n,n}^{q-2}(z),\qquad z\in{\mathbb{D}},$

where $R:=\big{(}1-2\big{(}2|z|^{2}-1\big{)}t+t^{2}\big{)}^{1/2}$, is a member
of $\Psi^{+}(\Omega_{2q})$, for each $t\in(0,1)$.

5\. Horn family. Let $r$, $R$ be positive integers such that $4r=(R-1)^{2}$.
Horn’s function $H_{4}$ is defined on p. 57 of Srivastava and Manocha [42] by

$\displaystyle
H_{4}(a,b;c,d;x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{2m+n}(b)_{n}}{(c)_{m}(d)_{n}}\frac{x^{m}y^{n}}{m!n!},$

where $|x|<r$ and $|y|<R$. An application of Theorem 2.2 in Aktaş et al. [2]
shows that

$\displaystyle
f_{t,s,b}(z):=\frac{1}{(1-s)^{q-1}}H_{4}\left(q-1,b;q-1,q-1;\frac{s(|z|^{2}-1)}{(1-s)^{2}},\frac{t\overline{z}}{1-s}\right)$
$\displaystyle\hphantom{f_{t,s,b}(z)}{}=\sum_{m,n\geq
0}(q+n-1)_{m}(b)_{n}\frac{t^{n}s^{m}}{m!n!}R_{m,m+n}^{q-2}(z),\qquad
z\in{\mathbb{D}}.$

Hence it is a member of $\Psi^{+}(\Omega_{2q})$, for each $b$, a positive
integer, and $t$, $s$ positive numbers satisfying

$\displaystyle|s|<1,\qquad\frac{|s|}{(1-s)^{2}}<r,\qquad\text{and}\qquad\frac{|t|}{1-s}<R.$

6\. Lauricella family. Let $r_{1}$, $r_{2}$ and $r_{3}$ be positive integers
such that $r_{1}r_{2}=(1-r_{2})(r_{2}-r_{3})$. The Lauricella hypergeometric
function of three variables $F_{14}$ (Saran’s notation $F_{F}$ is also used
(Saran [32])) is defined by (see p. 67 of Srivastava and Manocha [42])

$\displaystyle
F_{14}(a_{1},a_{1},a_{1},b_{1},b_{2},b_{1};c_{1},c_{2},c_{2};x_{1},x_{2},x_{3})=\sum_{m,n,p=0}^{\infty}\frac{(a_{1})_{m+n+p}(b_{1})_{m+p}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n+p}}\frac{x_{1}^{m}x_{2}^{n}x_{3}^{p}}{m!n!p!},$

where $|x_{1}|<r_{1}$, $|x_{2}|<r_{2}$ and $|x_{3}|<r_{3}$. For
$t,s\in\mathbb{R}$ such that $|s|<r_{1}$ and $|t|<r_{2}$, where
$r_{1}=r_{2}(1-r_{2})$, define

$\displaystyle
f_{t,s,b}(z):=F_{14}\big{(}1,1,1,q-1,b,q-1;q-1,1,1;s\big{(}|z|^{2}-1\big{)},tz,s|z|^{2}\big{)},\qquad
z\in{\mathbb{D}}.$

From Theorem 2.3 in Aktaş et al. [2] we get

$\displaystyle f_{t,s,b}(z)=\sum_{m,n\geq
0}(q-1)_{n}(b)_{m}\frac{t^{m}s^{n}}{m!n!}R_{m+n,n}^{q-2}(z),\qquad
z\in{\mathbb{D}},$

and hence, $f_{t,s,b}$ is a member of $\Psi^{+}(\Omega_{2q})$, for each $b$, a
positive integer, and $t$, $s$ positive numbers satisfying the relevant
conditions above.

Some comments are in order. Lauricella functions are generalizations of the
Gauss hypergeometric functions to multiple variables and were introduced by
Lauricella in 1893. Recursion formulas and integral representation for
Lauricella functions, including $F_{14}$ ($F_{F}$), have been studied and can
be found, for example, in Sahai and Verma [31] and Saran [33, 34]. In 1873,
Schwarz [41] found a list of 15 cases where hypergeometric functions can be
expressed algebraically. More precisely, Schwarz gave a list of parameters
determining the cases where the hypergeometric differential equation has two
independent solutions that are algebraic functions. Between 1989 and 2009
several researchers extended this list: to general one-variable hypergeometric
functions ${}_{p+1}F_{p}$ (Beukers and Heckman [8]), the Appell–Lauricella
functions $F_{1}$ and $F_{D}$ (Beazley Cohen and Wolfart [5]), the Appell
functions $F_{2}$ and $F_{4}$ (Kato [20, 21]), and the Horn function $G_{3}$
(Schipper [38]). In 2012, Bod [9] extended Schwarz’ list to the four classes
of Appell–Lauricella functions and the 14 complete Horn functions, including
$H_{4}$.

##  3 Proof of the results

In this section we first prove some technical lemmas. Then, we shall be able
to give the proof of our main results and to present the counterexamples.

The first lemma contains recurrence formulas connecting disc polynomials of
different indexes and degrees. They are obtained from equation (5.5) in
Aharmim et al. [1] and the following properties of the disc polynomials

$\displaystyle\overline{R_{m,n}^{\alpha}(z)}=R_{n,m}^{\alpha}(z),\qquad\overline{{\cal
D}_{z}R_{m,n}^{\alpha}(z)}={\cal
D}_{\overline{z}}R_{n,m}^{\alpha}(z),\qquad\alpha>-1,\quad m,n\geq 0,\quad
z\in{\mathbb{D}}.$

We observe that the normalization adopted in Aharmim et al. [1] for the disc
polynomials is different from the one we use here.

######  Lemma 3.1.

Let $m$, $n$ be non negative integers and $\alpha>-1$ be a real number. Then,
for any $z\in{\mathbb{D}}$, we have

$\displaystyle(\alpha+1)R_{m,n+1}^{\alpha}(z)=(\alpha+1)\overline{z}R_{m,n}^{\alpha+1}(z)-\big{(}1-|z|^{2}\big{)}{\cal
D}_{z}R_{m,n}^{\alpha+1}(z),$ (3.1)

and

$\displaystyle(\alpha+1)R_{n+1,m}^{\alpha}(z)=(\alpha+1){z}R_{n,m}^{\alpha+1}(z)-\big{(}1-|z|^{2}\big{)}{\cal
D}_{\overline{z}}R_{n,m}^{\alpha+1}(z).$ (3.2)

Below we prove an important technical result, that connects the expansion of a
continuously differentiable function $f$ in terms of the disc polynomials
$R_{m,n}^{\alpha}$ with the expansion of its derivatives in terms of the disc
polynomials $R_{m,n}^{\alpha+1}$.

Since $R_{m,n}^{q-2}$ belongs to $\Psi(\Omega_{2q})$ when $q\geq 2$ is an
integer, this connection will be the main ingredient in order to obtain
preservation of positive definiteness for the Descente operators, when walks
through dimensions over complex spheres are provided.

######  Lemma 3.2.

Let $f\colon{\mathbb{D}}\to\mathbb{C}$ be continuously differentiable and let
$\alpha>-1$ be a real number. Consider the expansion of $f$ in terms of the
disc polynomials $R_{m,n}^{\alpha}$ and the expansions of ${\cal D}_{z}f$ and
${\cal D}_{\overline{z}}f$ in terms of the disc polynomials
$R_{m,n}^{\alpha+1}$

$\displaystyle
f(z)\sim\sum_{m,n=0}^{\infty}a_{m,n}^{\alpha}R_{m,n}^{\alpha}(z),\qquad
z\in{\mathbb{D}},$ $\displaystyle{\cal
D}_{z}f(z)\sim\sum_{m,n=0}^{\infty}b_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z)\qquad\text{and}\qquad{\cal
D}_{\overline{z}}f(z)\sim\sum_{m,n=0}^{\infty}\tilde{b}_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z),\qquad
z\in{\mathbb{D}}.$

Then,

$\displaystyle
b_{m,n}^{\alpha+1}=\frac{(m+1)(n+\alpha+1)}{(\alpha+1)}a_{m+1,n}^{\alpha},\qquad
m,n\geq 0,$

and

$\displaystyle\tilde{b}_{m,n}^{\alpha+1}=\frac{(n+1)(m+\alpha+1)}{(\alpha+1)}a_{m,n+1}^{\alpha},\qquad
m,n\geq 0.$

It is worth noting that this result is not surprising if we consider the
identities obtained in Koornwinder [23]: for $\alpha>-1$,

$\displaystyle{\cal
D}_{z}R_{m,n}^{\alpha}=c_{\alpha}(m,n)R_{m-1,n}^{\alpha+1}\qquad\text{and}\qquad{\cal
D}_{\overline{z}}R_{m,n}^{\alpha}=c_{\alpha}(n,m)R_{m,n-1}^{\alpha+1},$ (3.3)

where $c_{\alpha}(m,n):=(m(n+\alpha+1))/(\alpha+1)$. These are, in the complex
case, the analogue of the identities for the derivative of the Gegenbauer
polynomials (see Szegő [44, equation (4.7.14)]).

Actually, Lemma 3.2 shows that the coefficients in the expansions are linked
as if the series could be derived term by term.

###### Proof of Lemma 3.2.

The coefficients $b_{m,n}^{\alpha+1}$ are given by the formula

$\displaystyle b_{m,n}^{\alpha+1}=h_{m,n}^{\alpha+1}\int_{{\mathbb{D}}}{\cal
D}_{z}f(z)\overline{R_{m,n}^{\alpha+1}(z)}{\rm d}\nu_{\alpha+1}(z),$

where the constants $h_{m,n}^{\alpha+1}$ are given in (2.6). Define

$\displaystyle I:=\int_{{\mathbb{D}}}{\cal
D}_{z}f(z){R_{n,m}^{\alpha+1}(z)}{\rm
d}\nu_{\alpha+1}(z)=\frac{\alpha+2}{\pi}\int_{{\mathbb{D}}}{\cal
D}_{z}f(z){R_{n,m}^{\alpha+1}(z)}\big{(}1-x^{2}-y^{2}\big{)}^{\alpha+1}{\rm
d}x{\rm d}y.$

Integration by parts and direct inspection shows that

$\displaystyle I=\frac{\alpha+2}{\pi}\bigg{\\{}\int_{{\mathbb{D}}}{\cal
D}_{z}\big{[}f(z){R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}\big{]}{\rm
d}x{\rm d}y$
$\displaystyle\hphantom{I=\frac{\alpha+2}{\pi}\bigg{\\{}}{}-\int_{{\mathbb{D}}}f(z){\cal
D}_{z}\big{[}{R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}\big{]}{\rm
d}x{\rm d}y\bigg{\\}}.$

Using Green’s theorem and (1.2) we have

$\displaystyle\int_{\Omega_{2}}g(z){\rm
d}\overline{z}=-2i\int_{{\mathbb{D}}}{\cal D}_{z}(g)(z){\rm d}x{\rm d}y,$

for any continuously differentiable function $g$. Thus,

$\displaystyle
I=\frac{\alpha+2}{\pi}\left\\{\frac{i}{2}\int_{\Omega_{2}}\\!f(z){R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}{\rm
d}\overline{z}-\\!\int_{{\mathbb{D}}}f(z){\cal
D}_{z}\left[{R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}\right]{\rm
d}x{\rm d}y\right\\}$
$\displaystyle\hphantom{I}{}=-\frac{\alpha+2}{\pi}\int_{{\mathbb{D}}}f(z){\cal
D}_{z}\left[{R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}\right]{\rm
d}x{\rm d}y.$

Now, by noting that

$\displaystyle{\cal
D}_{z}\big{[}{R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}\big{]}={\cal
D}_{z}{R_{n,m}^{\alpha+1}(z)}\big{(}1-|z|^{2}\big{)}^{\alpha+1}-(\alpha+1)\big{(}1-|z|^{2}\big{)}^{\alpha}\overline{z}{R_{n,m}^{\alpha+1}(z)},$

we get

$\displaystyle
I=\frac{\alpha+2}{\pi}\int_{{\mathbb{D}}}f(z)\big{(}1-|z|^{2}\big{)}^{\alpha}\left[(\alpha+1)\overline{z}R_{n,m}^{\alpha+1}(z)-\big{(}1-|z|^{2}\big{)}{\cal
D}_{z}{R_{n,m}^{\alpha+1}(z)}\right]{\rm d}x{\rm d}y.$

Hence, using Lemma 3.1, we have

$\displaystyle
I=\frac{\alpha+2}{\pi}\int_{{\mathbb{D}}}f(z)\big{(}1-|z|^{2}\big{)}^{\alpha}(\alpha+1)R_{n,m+1}^{\alpha}(z){\rm
d}x{\rm
d}y=(\alpha+2)\int_{{\mathbb{D}}}f(z)\overline{R_{m+1,n}^{\alpha}(z)}{\rm
d}\nu_{\alpha}(z).$

Thus,

$\displaystyle
b_{m,n}^{\alpha+1}=h_{m,n}^{\alpha+1}I=(\alpha+2)h_{m,n}^{\alpha+1}\frac{1}{h_{m+1,n}^{\alpha}}a_{m+1,n}^{\alpha}.$

Replacing the values of $h_{m,n}^{\alpha+1}$ and $h_{m+1,n}^{\alpha}$ given in
equation (2.6), we obtain

$\displaystyle
b_{m,n}^{\alpha+1}=(\alpha+2)\frac{(m+1)(\alpha+n+1)}{(\alpha+2)(\alpha+1)}a_{m+1,n}^{\alpha}=\frac{(m+1)(\alpha+n+1)}{(\alpha+1)}a_{m+1,n}^{\alpha}.$

The proof for the case of the operator ${\cal D}_{\overline{z}}$ is analogous
observing that

$\displaystyle\int_{\Omega_{2}}g(z){\rm d}z=2i\int_{{\mathbb{D}}}{\cal
D}_{\overline{z}}(g)(z){\rm d}x{\rm d}y.$ ∎

The last technical lemma gives a condition for the expansion of a continuous
function in terms of the disc polynomials to be uniformly convergent.

######  Lemma 3.3.

Let $g\colon{\mathbb{D}}\to\mathbb{C}$ be a continuous function and consider
its expansion

$\displaystyle g(z)\sim\sum_{m,n\geq 0}d_{m,n}^{q-2}R_{m,n}^{q-2}(z),\qquad
z\in{\mathbb{D}},$ (3.4)

where $d_{m,n}^{q-2}$ are given as in (2.12). If $d_{m,n}^{q-2}\geq 0$ for all
$m,n\geq 0$, then $\sum\limits_{m,n\geq 0}d_{m,n}^{q-2}<\infty$. In
particular, the series in (3.4) converges uniformly in ${\mathbb{D}}$.

###### Proof.

The argument is similar to the one used in the proof of Theorem 4.1 in
Menegatto and Peron [26]. Given $\xi\in\Omega_{2q}$, consider the continuous
function $h(\rho):=g(\langle\rho,\xi\rangle)$, $\rho\in\Omega_{2q}$. By
equation (2.11), the solution of the Dirichlet problem $\Delta_{2q}u=0$ in the
interior of $B_{2q}$ with boundary condition $h$, evaluated on the segment
$r\xi$, $r\in[0,1)$, is

$\displaystyle u(r\xi)=\int_{\Omega_{2q}}{\cal
P}_{q}(r\xi,\rho)g(\langle\rho,\xi\rangle){\rm
d}\omega_{2q}(\rho)=\sum_{m,n\geq 0}S_{m,n}^{q}(r)d_{m,n}^{q-2},$

where the last equality is obtained from (2.10), (2.12).

Since $u$ is continuous up to the boundary and coincides with $h$ on
$\Omega_{2q}$, we obtain

$\displaystyle\lim_{r\to 1^{-}}\sum_{m,n\geq
0}d_{m,n}^{q-2}S_{m,n}^{q}(r)=\lim_{r\to
1^{-}}u(r\xi)=u(\xi)=g(\langle\xi,\xi\rangle)=g(1).$

Now, note that

$\displaystyle
0\leq\sum_{m=0}^{k}\sum_{n=0}^{l}d_{m,n}^{q-2}S_{m,n}^{q}(r)\leq\sum_{m,n\geq
0}d_{m,n}^{q-2}S_{m,n}^{q}(r),\qquad 0\leq r<1.$

Letting $r\to 1^{-}$, we get

$\displaystyle 0\leq
s_{k,l}:=\sum_{m=0}^{k}\sum_{n=0}^{l}d_{m,n}^{q-2}\leq\lim_{r\to
1^{-}}\sum_{m,n\geq 0}d_{m,n}^{q-2}S_{m,n}^{q}(r)=g(1),\qquad
k,l\in\mathbb{Z}_{+}.$

Hence, the sequence $\\{s_{k,l}\\}_{k,l\in\mathbb{Z}_{+}}$ is bounded and
increasing. Thus, the series $\sum\limits_{m,n\geq 0}d_{m,n}^{q-2}$ is
convergent. Using the fact that $|R_{m,n}^{q-2}(z)|\leq 1$ for all
$z\in{\mathbb{D}}$ and using the Weierstrass M-Test, the proof is completed. ∎

At this point, we are able to prove our main results.

###### Proof of Theorem 1.1..

Let $f$ be a function in the class $\Psi(\Omega_{2q})$. Then, by Theorem
2.1(1),

$\displaystyle f(z)=\sum_{m,n\geq 0}a_{m,n}^{\alpha}R_{m,n}^{\alpha}(z),\qquad
z\in{\mathbb{D}},$

where $\alpha=q-2$, $a_{m,n}^{\alpha}\geq 0$, for all $m,n\geq 0$, and
$\sum\limits_{m,n\geq 0}a_{m,n}^{\alpha}<\infty$. Consider the expansion in
terms of disc polynomials of ${\cal D}_{z}f$:

$\displaystyle{\cal D}_{z}f(z)\sim\sum_{m,n\geq
0}b_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z),\qquad z\in{\mathbb{D}}.$

By Lemma 3.2 and equation (3.3),

$\displaystyle b_{m,n}^{\alpha+1}=c_{\alpha}(m+1,n)a_{m+1,n}^{\alpha},\qquad
m,n\geq 0.$ (3.5)

Roughly speaking, (3.5) means that the coefficients
$\big{\\{}b_{m,n}^{\alpha+1}\big{\\}}$ are obtained from the
$\\{a_{m,n}^{\alpha}\\}$ by suppressing the $a_{0,n}^{\alpha}$, translating in
the first index and multiplying by the positive constants
$\\{c_{\alpha}(m+1,n)\\}$.

Then, by equation (3.3), we have

$\displaystyle\sum_{m,n\geq 0}a_{m,n}^{\alpha}{\cal
D}_{z}R_{m,n}^{\alpha}(z)=\sum_{m\geq-1}\sum_{n\geq 0}a_{m+1,n}^{\alpha}{\cal
D}_{z}R_{m+1,n}^{\alpha}(z)=\sum_{m,n\geq
0}b_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z).$

Now, since $c_{\alpha}(m+1,n)$ are positive constants, we have that
$b_{m,n}^{\alpha+1}\geq 0$ for all $m,n\geq 0$. By Lemma 3.3, the series

$\displaystyle\sum_{m,n\geq 0}a_{m,n}^{\alpha}{\cal
D}_{z}R_{m,n}^{\alpha}(1)=\sum_{m,n\geq 0}b_{m,n}^{\alpha+1}$

is convergent and the series $\sum\limits_{m,n\geq
0}b_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z)$ converges uniformly in
${\mathbb{D}}$. It follows, by term by term differentiation, that

$\displaystyle{\cal D}_{z}f(z)=\sum_{m,n\geq
0}b_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z).$

Hence, by Theorem 2.1(1), ${\cal D}_{z}f$ belongs to the class
$\Psi(\Omega_{2q+2})$.

Similarly, we can conclude the same for the operator ${\cal
D}_{\overline{z}}$.

For the item (ii), observe that, as a consequence of (3.5), if the
$(2q)$-complex Schoenberg coefficients $a_{m,n}^{\alpha}$ of $f$ satisfy
(1.4), then the $(2q+2)$-complex Schoenberg coefficients $b_{m,n}^{\alpha+1}$
of ${\cal D}_{z}f$ (and similarly for ${\cal D}_{\overline{z}}f$) satisfy
(2.14). Actually, the condition $m,n\geq 1$ in the set considered in (1.4)
guarantees that the intersections with the arithmetic progressions in
$\mathbb{Z}$ do not depend on the coefficients $a_{m,0}^{\alpha}$ or
$a_{0,n}^{\alpha}$, which are suppressed by the Descente operators.

The results for ${\cal D}_{x}f$ follow immediately by (1.2). ∎

###### Proof of Theorem 1.2.

Suppose that $f$ belongs to the class $\Psi(\Omega_{2q+2})$. By Theorem 2.1,

$\displaystyle f(z)=\sum_{m,n\geq
0}a_{m,n}^{\alpha+1}R_{m,n}^{\alpha+1}(z),\qquad z\in{\mathbb{D}},$

where $\alpha=q-2$ and $a_{m,n}^{\alpha+1}\geq 0$ for all $m,n\geq 0$ and
$\sum\limits_{m,n\geq 0}a_{m,n}^{\alpha+1}<\infty$. By equation (3.3), we have

$\displaystyle{\cal
I}(R_{m-1,n}^{\alpha+1})(z)=\frac{1}{c_{\alpha}(m,n)}\big{(}R_{m,n}^{\alpha}(z)-R_{m,n}^{\alpha}(0)\big{)},$
(3.6)

where $R_{n,n}^{\alpha}(0)=(-1)^{n}n!\alpha!/(n+\alpha)!$ and
$R_{m,n}^{\alpha}(0)=0$, $m\neq n$ (Wünsche [46, equation (2.9)]). Thus
consider

$\displaystyle F(z):=\sum_{m,n\geq 0}a_{m,n}^{\alpha+1}{\cal
I}(R_{m,n}^{\alpha+1})(z)=\sum_{m\geq 1}\sum_{n\geq
0}\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)}\left(R_{m,n}^{\alpha}(z)-R_{m,n}^{\alpha}(0)\right),\qquad
z\in{\mathbb{D}}.\\!\\!\\!$ (3.7)

Since $c_{\alpha}(m,n)\geq 1$ for all $m\geq 1$, $n\geq 0$ and
$|R_{m,n}^{\alpha}(0)|\leq 1$, for all $m$, $n$, we have that the series

$\displaystyle\sum_{m\geq 1}\sum_{n\geq
0}\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)}\qquad\text{and}\qquad
c:=\sum_{m\geq 1}\sum_{n\geq
0}\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)}R_{m,n}^{\alpha}(0)$

are convergent. Furthermore, since

$\displaystyle\left|\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)}\left(R_{m,n}^{\alpha}(z)-R_{m,n}^{\alpha}(0)\right)\right|\leq
2\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)},\qquad m,n\geq 0,\quad
z\in{\mathbb{D}},$

the series in (3.7) converges uniformly in ${\mathbb{D}}$. On the other hand,
by applying the derivation operator ${\cal D}_{z}$ term by term in (3.7), one
obtains the uniformly convergent series of $f$. Then $F$ is a $z$-primitive of
$f$. Since $F(0)$=0, we conclude that (3.7) converges to ${\cal I}(f)(z)$.

We can now write

$\displaystyle{\cal I}(f)(z)=\sum_{m,n\geq
0}b_{m,n}^{\alpha}R_{m,n}^{\alpha}(z),$

where

$\displaystyle b_{0,0}^{\alpha}:=-c;\qquad b_{0,n}^{\alpha}:=0,\qquad n\geq
1;$ $\displaystyle
b_{m,n}^{\alpha}:=\frac{a_{m-1,n}^{\alpha+1}}{c_{\alpha}(m,n)},\qquad m\geq
1,\quad n\geq 0;$ (3.8)

and $\sum\limits_{m,n\geq 0}b_{m,n}^{\alpha}<\infty$. Now we can write

$\displaystyle c+{\cal I}f(z)=\sum_{m,n\geq
0}\widehat{b}_{m,n}^{\alpha}R_{m,n}^{\alpha}(z),$

where

$\displaystyle\widehat{b}_{0,0}^{\alpha}:=0;\qquad\widehat{b}_{0,n}^{\alpha}:=b_{0,n}^{\alpha},\qquad
n\geq
1\qquad\text{and}\qquad\widehat{b}_{m,n}^{\alpha}:=b_{m,n}^{\alpha},\qquad
m\geq 1,\quad n\geq 0$ (3.9)

are nonnegative constants and $\sum\limits_{m,n\geq
0}\widehat{b}_{m,n}^{\alpha}<\infty$.

Equations (3.8) and (3.9) mean that the coefficients
$\big{\\{}\widehat{b}_{m,n}^{\alpha}\big{\\}}$ are obtained from the
$\\{a_{m,n}^{\alpha+1}\\}$, by translating in the first index, adding the new
coefficients $\widehat{b}_{0,n}^{\alpha}=0$, and dividing by the positive
constants $\\{c_{\alpha}(m,n)\\}$.

Hence, applying Theorem 2.1(1) again, we have that $c+{\cal I}f$ belongs to
the class $\Psi(\Omega_{2q})$.

For the item (ii), it is enough to observe that the $(2q+2)$-complex
Schoenberg coefficients $a_{m,n}^{\alpha+1}$ of $f$ satisfy (2.14) by the
assumption $f\in\Psi^{+}(\Omega_{2q+2})$, then, as a consequence of (3.8),
(3.9), also the $(2q)$-complex Schoenberg coefficients
$\widehat{b}_{m,n}^{\alpha}$ of $c+{\cal I}f$ satisfy (2.14), implying
$c+{\cal I}f\in\Psi^{+}(\Omega_{2q})$.

For the operator ${\overline{\cal I}}$, one uses the relation

$\displaystyle{\overline{\cal
I}}(R_{m,n-1}^{\alpha+1})(z)=\frac{1}{c_{\alpha}(n,m)}\big{(}R_{m,n}^{\alpha}(z)-R_{m,n}^{\alpha}(0)\big{)},$

and follows the same arguments. In fact, the $(2q)$-complex Schoenberg
coefficients of $C+{\overline{\cal I}}f$ are given by

$\displaystyle\check{b}_{0,0}^{\alpha}:=C-\sum_{\mu\geq 1}\sum_{\nu\geq
0}\frac{a_{\mu,\nu-1}^{\alpha+1}}{c_{\alpha}(\nu,\mu)}R_{\mu,\nu}^{\alpha}(0);\qquad\check{b}_{m,0}^{\alpha}:=0,\qquad
m\geq 1;$
$\displaystyle\check{b}_{m,n}^{\alpha}:=\frac{a_{m,n-1}^{\alpha+1}}{c_{\alpha}(m,n)},\qquad
m\geq 0,\quad n\geq 1.$ ∎

###### Proof of Counterexample 1.3..

Let us denote by $a_{m,n}^{q-2}(g)$ the $(2q)$-complex Schoenberg coefficients
of a positive definite function $g$. Theorem 2.1(2) is required.

(i) For a function $f$ as in the statement, we have ${\cal D}_{x}f={\cal
D}_{z}f$ and

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-1}({\cal
D}_{z}f)>0\big{\\}}=\big{\\{}m-n\colon
a_{m,n}^{q-2}(f)>0\big{\\}}=\mathbb{Z}_{+}.$

Hence the above set intercepts every arithmetic progression in $\mathbb{Z}$,
that is $f\in\Psi^{+}(\Omega_{2q})$ and ${\cal
D}_{z}f,D_{x}f\in\Psi^{+}(\Omega_{2q+2})$. However, ${\cal
D}_{\overline{z}}f\equiv 0$, so that ${\cal
D}_{\overline{z}}f\not\in\Psi^{+}(\Omega_{2q+2})$.

(ii) Analogous to (i).

(iii) For a function $f$ as in the statement, we have

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-2}(f)>0,\,m,n\geq
0\big{\\}}=\left(\bigcup_{j=2}^{5}5\mathbb{Z}_{+}+j\right)\cup(-5\mathbb{Z}_{+}-4),$

which intercepts every arithmetic progression in $\mathbb{Z}$ and then
$f\in\Psi^{+}(\Omega_{2q})$. However

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-1}(\mathcal{D}_{z}f)>0,\,m,n\geq
0\big{\\}}=\mathbb{Z}_{+}\setminus 5\mathbb{Z}$

and

$\displaystyle\big{\\{}m-n\colon a_{m,n}^{q-1}({\cal
D}_{\overline{z}}f)>0,\,m,n\geq 0\big{\\}}=-5\mathbb{Z}_{+}-3,$

that is, ${\cal D}_{z}f,{\cal
D}_{\overline{z}}f\not\in\Psi^{+}(\Omega_{2q+2})$. To see that
$\mathcal{D}_{x}f\not\in\Psi(\Omega_{2q+2})$, note that $\\{m-n\colon
a_{m,n}^{q-1}(\mathcal{D}_{x}f)>0$, $m,n\geq 0\\}$ is the union of the
previous two sets, so it does not intersect the progression $5\mathbb{Z}$. ∎

### Acknowledgement

The authors gratefully thank the anonymous referees for the constructive
comments and recommendations which helped to greatly improve the paper.
Eugenio Massa was supported by grant $\\#$2014/25398-0, São Paulo Research
Foundation (FAPESP) and grant $\\#$308354/2014-1, CNPq/Brazil. Ana P. Peron
was supported by grants $\\#$2016/03015-7 and $\\#$2014/25796-5, São Paulo
Research Foundation (FAPESP). Emilio Porcu was supported by grant FONDECYT
#1170290 from the Chilean government.

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