# Master Thesis
On the almost everywhere and norm convergences of Nörlund means with respect
to Vilenkin systems

Zura Dvalashvili and Nato Nadirashvili
The University of Georgia
School of Science and Technology
77a Merab Kostava St, Tbilisi, 0128, Georgia
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>
Supervisor:
George Tephnadze
The University of Georgia
School of Science and Technology
77a Merab Kostava St, Tbilisi, 0128, Georgia
e-mail<EMAIL_ADDRESS>
(2021)

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###### Contents

  1. Preface
  2. Key words
  3. 1 Introduction
    1. 1.1 Vilenkin Groups and Functions
    2. 1.2 $L_{p}$ and weak-$L_{p}$ Spaces
    3. 1.3 Dirichlet and Vilenkin-Fejér Kernels
  4. 2 Nörlund Means of Vilenkin-Fourier series in Lebesgue Spaces
    1. 2.1 Introduction
    2. 2.2 Well-known and New examples of Nörlund Means
    3. 2.3 Kernels of Nörlund Means
    4. 2.4 Norm Convergence of Nörlund Means in Lebesgue Spaces
    5. 2.5 Convergence of Nörlund Means in Vilenkin-Lebesgue points

### Preface

Unlike the classical theory of Fourier series which deals with decomposition
of a function into sinusoidal waves the Vilenkin (Walsh) functions are
rectangular waves. The development of the theory of Vilenkin-Fourier series
has been strongly influenced by the classical theory of trigonometric series
but there are a lot of differences also. The aim of my master thesis is to
discuss, develop and apply the newest developments of this fascinating theory
connected to modern harmonic analysis. In particular, we investigate Nörlund
means but only in the case when their coefficients are monotone and prove
convergence in Lebesgue and Vilenkin-Lebesgue points. Since almost everywhere
points are Lebesgue and Vilenkin-Lebesgue points for any integrable functions
we obtain almost everywhere convergence of such summability methods.

### Key words

Key words: Vilenkin systems, Nörlund means, almost everywhere convergence,
Lebesgue points, Vilenkin-Lebesgue points, convergence, approximation.

##  Chapter 1 Introduction

###  1.1 Vilenkin Groups and Functions

Denote by $\mathbb{N}_{+}$ the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}$, $\mathbb{Z}$ the set of the integers,
$\mathbb{R}$ the real numbers, $\mathbb{R}_{+}$ the positive real numbers,
$\mathbb{C}$ the complex numbers. Let $m:=(m_{0},m_{1},\ldots)$ be a sequence
of positive integers not less than 2. Denote by

$Z_{m_{k}}:=\\{0,1,\ldots,m_{k}-1\\}$

the additive group of integers modulo $m_{k}$ .

Define the group $G_{m}$ as the complete direct product of the groups
$Z_{m_{k}}$ with the product of the discrete topologies of $Z_{m_{k}}$.

The direct product $\mu$ of the measures

$\mu_{k}\left(j\right):=1/m_{k}\qquad(j\in Z_{m_{k}})$

is the Haar measure on $G_{m}$ with $\mu\left(G_{m}\right)=1.$

If $\sup_{n\in\mathbb{N}}m_{n}<\infty$, then we call $G_{m}$ a bounded
Vilenkin group. If the generating sequence $m$ is not bounded, then $G_{m}$ is
said to be an unbounded Vilenkin group.

In this book we discuss only bounded Vilenkin groups, i.e. the case when
$\sup_{n\in\mathbb{N}}m_{n}<\infty.$

The elements of $G_{m}$ are represented by sequences

$x:=\left(x_{0},x_{1},\ldots,x_{j},\ldots\right)\qquad\left(x_{j}\in
Z_{m_{j}}\right).$

It is easy to give a base for the neighborhoods of $G_{m}:$

$\displaystyle I_{0}\left(x\right)$ $\displaystyle:$ $\displaystyle=G_{m},$
$\displaystyle I_{n}(x)$ $\displaystyle:$ $\displaystyle=\\{y\in G_{m}\mid
y_{0}=x_{0},\ldots,y_{n-1}=x_{n-1}\\}\qquad\left(x\in G_{m},\ \
n\in\mathbb{N}\right).$

We call subsets $I_{n}(x)\subset G_{m}$ Vilenkin intervals. Let

$e_{n}:=\left(0,\ldots,0,x_{n}=1,0,\ldots\right)\in
G_{m}\qquad\left(n\in\mathbb{N}\right).$

If we define $I_{n}:=I_{n}\left(0\right),$ for $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G_{m}\ \ \backslash$ $I_{n},$ then

$\overline{I_{N}}=\overset{N-1}{\underset{s=0}{\bigcup}}I_{s}\backslash
I_{s+1}=\left(\overset{N-2}{\underset{k=0}{\bigcup}}\overset{N-1}{\underset{l=k+1}{\bigcup}}I_{N}^{k,l}\right)\bigcup\left(\underset{k=1}{\bigcup\limits^{N-1}}I_{N}^{k,N}\right),$
(1.1.1)

where

$I_{N}^{k,l}:=\left\\{\begin{array}[]{l}I_{N}(0,\ldots,0,x_{k}\neq
0,0,...,0,x_{l}\neq 0,x_{l+1},\ldots,x_{N-1},\ldots),\\\ \text{for}\qquad
k<l<N,\\\ I_{N}(0,\ldots,0,x_{k}\neq
0,x_{k+1}=0,\ldots,x_{N-1}=0,x_{N},\ldots),\\\ \text{for }\qquad
l=N.\end{array}\right.$

If we define the so-called generalized number system based on $m$ in the
following way :

$M_{0}:=1,\ \ M_{k+1}:=m_{k}M_{k}\ \ (k\in\mathbb{N}),$

then every $n\in\mathbb{N}$ can be uniquely expressed as

$n=\sum_{j=0}^{\infty}n_{j}M_{j},$

where $n_{j}\in Z_{m_{j}}\ \ (j\in\mathbb{N}_{+})$ and only a finite number of
$n_{j}^{{}^{\prime}}$‘s differ from zero.

The Vilenkin group can be metrizable with the following metric:

$\rho\left(x,y\right):=\left|x-y\right|:=\sum_{k=0}^{\infty}\frac{\left|x_{k}-y_{k}\right|}{M_{k+1}},\qquad\left(x,y\in
G_{m}\right).$

For the natural numbers $n=\sum_{j=1}^{\infty}n_{j}M_{j}$ and
$k=\sum_{j=1}^{\infty}k_{j}M_{j}$ we define

$n\widehat{+}k:=\sum_{i=0}^{\infty}{\left(n_{i}\oplus k_{i}\right)}{M_{i+1}}$

and

$n\widehat{-}k:=\sum_{i=0}^{\infty}{\left(n_{i}\ominus
k_{i}\right)}{M_{i+1}},$

where

$a_{i}\oplus b_{i}:=(a_{i}+b_{i})\text{mod}m_{i},\qquad a_{i},b_{i}\in
Z_{m_{i}}$

and $\ominus$ is the inverse operation for $\oplus$.

Next, we introduce on $G_{m}$ an orthonormal system, which is called Vilenkin
system (see [91, 92, 93]).

At first, we define the complex-valued function
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions, by

$r_{k}\left(x\right):=\exp\left(2\pi\imath x_{k}/m_{k}\right),\ \
\left(\imath^{2}=-1,\ \ x\in G_{m},\ \ k\in\mathbb{N}\right).$ (1.1.2)

Now, define the Vilenkin systems $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
by:

$\psi_{n}(x):=\prod\limits_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right),\ \
\left(n\in\mathbb{N}\right).$ (1.1.3)

Specifically, we call this system the Walsh-Paley system when $m\equiv 2.$

###### Proposition 1.1.1

(see [1]) Let $n\in\mathbb{N}.$ Then

$\displaystyle\left|\psi_{n}\left(x\right)\right|$ $\displaystyle=$
$\displaystyle 1,\text{ }$ $\displaystyle\psi_{n}\left(x-y\right)$
$\displaystyle=$ $\displaystyle\psi_{n}\left(x\right)\text{\
}\overline{\psi}_{n}\left(y\right).$

The direct product $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \qquad}(j\in Z_{m_{k}})$

is the Haar measure on $G_{m_{\text{ }}}$with $\mu\left(G_{m}\right)=1.$
Translation of a subset $I_{n}(x)\in G_{m}$ by $y$ is defined by
$\tau_{y}\left(I_{n}(x)\right)=\\{I_{n}(x)+y\\}.$ Since $\mu$ is a product
measure we get that $\mu\left(I_{n}(x)\right)=1/{M_{n}}$ and

$\mu\left(\tau_{y}\left(I_{n}(x)\right)\right)=\mu\left(I_{n}(x+y)\right)=1/M_{n}$

for all $y\in G_{m}.$ Hence,

$\mu\left(\tau_{y}\left(I_{n}(x)\right)\right)=\mu\left(I_{n}(x)\right).$

###### Proposition 1.1.2

Let $n,k\in\mathbb{N}.$ Then

$\int_{G_{m}}\psi_{n}d\mu\text{\thinspace}=\left\\{\begin{array}[]{ll}1\text{
\ \ \ \ }&n=0,\\\ 0&n\neq 0.\end{array}\right.$

Moreover, the Vilenkin systems are orthonormal, that is,

$\int_{G_{m}}\psi_{n}\overline{\psi_{k}}d\mu\text{\thinspace}=\left\\{\begin{array}[]{ll}1\text{
\ \ \ \ }&n=k,\\\ 0&n\neq k.\end{array}\right.$

###  1.2 $L_{p}$ and weak-$L_{p}$ Spaces

By a Vilenkin polynomial we mean a finite linear combination of Vilenkin
functions. We denote the collection of Vilenkin polynomials by $\mathcal{P}$.

Let $L^{0}(G_{m})$ represent the collection of functions which are almost
everywhere limits with respect to a measure $\mu$ of sequences in
$\mathcal{P}$.

For $0<p<\infty$ let $L^{p}(G_{m})$ represent the collection of $f\in
L^{0}(G_{m})$ such that

$\left\|f\right\|_{p}:=\left(\int_{G_{m}}\left|f\right|^{p}d\mu\right)^{1/p}$

is finite.

Denote by $L^{\infty}(G_{m})$ the space of all $f\in L^{0}(G_{m})$ for which

$\left\|f\right\|_{\infty}:=\inf\left\\{C>0:\mu\left\\{x\in
G_{m}:\left|f\right|>C\right\\}=0\right\\}<+\infty.$

The space $C(G_{m})$ consist all continuous function for which

$\left\|f\right\|_{C}:=\underset{x\in G_{m}}{\sup}\left|f(x)\right|<c<\infty.$

###### Proposition 1.2.1

(see [88]) Well-known Minkowski’s integral inequality is given by

$\left\|\int_{G_{m}}f\left(\cdot,t\right)dt\right\|_{p}\leq\int_{G_{m}}\left\|f\left(\cdot,t\right)\right\|_{p}dt,\
\ \ \text{for all}\ \ \ p\geq 1.$

The convolution of two functions $f,g\in L^{1}(G_{m})$ is defined by

$\left(f\ast
g\right)\left(x\right):=\int_{G_{m}}f\left(x-t\right)g\left(t\right)dt\text{ \
\ }\left(x\in G_{m}\right).$

It is easy to see that

$\left(f\ast
g\right)\left(x\right)=\int_{G_{m}}f\left(t\right)g\left(x-t\right)dt\text{ \
\ }\left(x\in G_{m}\right).$

###### Proposition 1.2.2

Let $f\in L^{r}\left(G_{m}\right),$ $g\in L^{1}\left(G_{m}\right)$ and $1\leq
r<\infty.$ Then $f\ast g\in L^{r}\left(G_{m}\right)$ and

$\left\|f\ast g\right\|_{r}\leq\left\|f\right\|_{r}\left\|g\right\|_{1}.$

First we present the following very important proposition:

###### Proposition 1.2.3

Since the Vilenkin function $\psi_{m}$ is constant on $I_{n}(x)$ for every
$x\in G_{m}$ and $0\leq m<M_{n},$ it is clear that each Vilenkin function is a
complex-valued step function, that is, it is a finite linear combination of
the characteristic functions. On the other hand, notice that, by Lemma 1.3.3
(Paley’s Lemma), it yields that

$\chi_{I_{n}(t)}\left(x\right)=\frac{1}{M_{n}}\sum_{j=0}^{M_{n}-1}\psi_{j}\left(x-t\right),\text{
\ \ }x\in I_{n}(t),$

for each $x,t\in$ $G_{m}$ and $n\in\mathbb{N}$. Thus each step function is a
Vilenkin polynomial. Consequently, we obtain that the collection of step
functions coincides with a collection of Vilenkin polynomials $\mathcal{P}$.

Since the Lebesgue measure is regular it follows that given $f\in L^{1}$ there
exist Vilenkin polynomials

$P_{1},P_{2}...,\ \ \ \text{ such that }\ \ \ P_{n}\rightarrow f\ \ \ \text{
a.e., as }\ \ \ n\rightarrow\infty.$

Moreover, any $f\in L^{p}(G_{m})$ can be written in the form $f=g-h$ where the
functions $g,\ h$ are almost everywhere limits of increasing sequences of non-
negative Vilenkin polynomials. In particular, $\mathcal{P}$ is dense in the
space $L^{p},$ for all $p\geq 1.$

The space $weak-L^{p}\left(G_{m}\right)$ consists of all measurable functions
$f$, for which

$\left\|f\right\|_{weak-L^{p}}:=\underset{y>0}{\sup}y\\{\mu\left(f>y\right)\\}^{1/p}<+\infty.$

###### Proposition 1.2.4

(see [88]) If $0<p\leq\infty,$ then
$L^{p}\left(G_{m}\right)\subset\text{weak}-L^{p}(G_{m})$ and

$\left\|f\right\|_{\text{weak}-L^{p}}\leq\left\|f\right\|_{p}.$

  * Proof. It is easy to see that

$\int_{G_{m}}\left|f\left(x\right)\right|^{p}dx\geq\int_{\left\\{x:\left|f\left(x\right)\right|>y\right\\}}\left|f\left(x\right)\right|^{p}dx\geq
y^{p}\mu\left(\left|f\right|>y\right),$

which proves the proposition.

An operator $T$ which maps a linear space of measurable functions on $G_{m}$
in the collection of measurable functions on $G_{m}$ is called sublinear if

$|T(f+g)|\leq|T(f)|+|T(g)|\text{ \ \ a.e. on \ \ }G_{m}\text{ \ \ and \ \
}|T(\alpha f)|=|\alpha||T(f)|$

for all scalars $\alpha$ and all $f$ in the domain of $T.$

###  1.3 Dirichlet and Vilenkin-Fejér Kernels

If $f\in L^{1}\left(G_{m}\right)$ we can define the Fourier coefficients, the
partial sums of Vilenkin-Fourier series, the Dirichlet kernels with respect to
Vilenkin systems in the usual manner:

$\displaystyle\widehat{f}\left(n\right)$ $\displaystyle:=$
$\displaystyle\int_{G_{m}}f\overline{\psi}_{n}d\mu,\qquad\left(n\in\mathbb{N}\right),$
$\displaystyle S_{n}f$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k},\qquad\left(n\in\mathbb{N}_{+}\right),$
$\displaystyle D_{n}$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{n-1}\psi_{k},\qquad\left(n\in\mathbb{N}_{+}\right),$

respectively. It is easy to see that

$\displaystyle S_{n}f\left(x\right)$ $\displaystyle=$
$\displaystyle\int_{G_{m}}f\left(t\right)\sum_{k=0}^{n-1}\psi_{k}\left(x-t\right)d\mu\left(t\right)$
$\displaystyle=$
$\displaystyle\int_{G_{m}}f\left(t\right)D_{n}\left(x-t\right)d\mu\left(t\right)=\left(f\ast
D_{n}\right)\left(x\right).$

The next well-known identities with respect to Dirichlet kernels (see Lemmas
1.3.1 and 1.3.2, Lemma 1.3.3) will be used many times in the proofs of our
main results:

###### Lemma 1.3.1

(see [1]) Let $n\in\mathbb{N}.$ Then

$D_{j+M_{n}}=D_{M_{n}}+\psi_{M_{n}}D_{j}=D_{M_{n}}+r_{n}D_{j},\qquad
j\leq\left(m_{n}-1\right)M_{n}$ (1.3.1)

and

$\displaystyle D_{M_{n}-j}(x)$ $\displaystyle=$ $\displaystyle
D_{M_{n}}(x)-\overline{\psi}_{M_{n}-1}(-x)D_{j}(-x)$ $\displaystyle=$
$\displaystyle D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{D}_{j}(x),\qquad
j<M_{n}.$

###### Lemma 1.3.2

(see [1]) Let $n\in\mathbb{N}$ and $1\leq s_{n}\leq m_{n}-1.$ Then

$D_{s_{n}M_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}\psi_{kM_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}r_{n}^{k}$
(1.3.3)

and

$D_{n}=\psi_{n}\left(\sum_{j=0}^{\infty}D_{M_{j}}\sum_{k=m_{j}-n_{j}}^{m_{j}-1}r_{j}^{k}\right),$
(1.3.4)

for $n=\sum_{i=0}^{\infty}n_{i}M_{i}.$

###### Lemma 1.3.3

(see [1] and [31]) (Paley’s Lemma) Let $n\in\mathbb{N}.$ Then

$D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{ll}M_{n},&x\in I_{n},\\\
0,&x\notin I_{n}.\end{array}\right.$

We also need the following estimate:

###### Lemma 1.3.4

(see [1])Let $n\in\mathbb{N}$. Then

$\|D_{M_{n}}\|_{1}=1.$

###### Lemma 1.3.5

(see [1] and [31]) Let $x\in I_{s}\backslash I_{s+1},\ \ s=0,...,N-1.$ Then

$\left|D_{n}\left(x\right)\right|\leq cM_{s}$

and

$\int_{I_{N}}\left|D_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\frac{cM_{s}}{M_{N}},$

where $c$ is an absolute constant.

It is obvious that

$\displaystyle\sigma_{n}f\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\overset{n-1}{\underset{k=0}{\sum}}\left(D_{k}\ast
f\right)\left(x\right)$ $\displaystyle=$ $\displaystyle\left(f\ast
K_{n}\right)\left(x\right)=\int_{G_{m}}f\left(t\right)K_{n}\left(x-t\right)d\mu\left(t\right),$

where $K_{n}$ are the so called Fejér kernels:

$\displaystyle K_{n}:=\frac{1}{n}\overset{n-1}{\underset{k=0}{\sum}}D_{k}.$

Using Abel transformation we get another representation of Fejér means

$\sigma_{n}f\left(x\right)=\overset{n-2}{\underset{k=0}{\sum}}\left(1-\frac{k}{n}\right)\widehat{f}\left(k\right)\psi_{k}\left(x\right)$

We frequently use the following well-known result:

###### Lemma 1.3.6

(see [22]) Let $n>t,$ $t,n\in\mathbb{N}.$ Then

$K_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{ll}\frac{M_{t}}{1-r_{t}\left(x\right)},&x\in
I_{t}\backslash I_{t+1},\qquad x-x_{t}e_{t}\in I_{n},\\\
\frac{M_{n}+1}{2},&x\in I_{n},\\\ 0,&\text{otherwise. }\end{array}\right.$

The proof of the next lemma can easily be done by using Lemma 1.3.6:

###### Lemma 1.3.7

Let $n\in\mathbb{N}$ and $x\in I_{N}^{k,l},$ where $k<l.$ Then

$K_{M_{n}}\left(x\right)=0,\text{ \ if \ }n>l,$ (1.3.5)

$\left|K_{M_{n}}\left(x\right)\right|\leq cM_{k},$ (1.3.6)

and

$\left|K_{M_{n}}(x)\right|\leq
c\sum_{s=0}^{n}M_{s}\sum_{r_{s}=1}^{m_{s}-1}\chi_{I_{n}(x-r_{s}e_{s})}$
(1.3.7)

Moreover,

$\int_{G_{m}}\left|K_{M_{n}}\right|d\mu\leq c<\infty,$ (1.3.8)

where $c$ is an absolute constant.

We also need the following useful result:

###### Lemma 1.3.8

(see [1] and [31]) Let $t,s_{n},\ \ n\in\mathbb{N},$ and $1\leq s_{n}\leq
m_{n}-1$. Then

$s_{n}M_{n}K_{s_{n}M_{n}}=\sum_{l=0}^{s_{n}-1}\left(\sum_{i=0}^{l-1}r_{n}^{i}\right)M_{n}D_{M_{n}}+\left(\sum_{l=0}^{s_{n}-1}r_{n}^{l}\right)M_{n}K_{M_{n}}$
(1.3.9)

The next equality for Fejér kernels is very important for our further
investigations:

###### Lemma 1.3.9

(see [1] and [31]) Let $n=\sum_{i=1}^{r}s_{n_{i}}M_{n_{i}}$, where
$n_{1}>n_{2}>\dots>n_{r}\geq 0$ and $1\leq s_{n_{i}}<m_{n_{i}}$ for all $1\leq
i\leq r$ as well as $n^{(k)}=n-\sum_{i=1}^{k}s_{n_{i}}M_{n_{i}}$, where
$0<k\leq r$. Then

$\displaystyle nK_{n}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{r}\left(\prod_{j=1}^{k-1}r_{n_{j}}^{s_{n_{j}}}\right)s_{n_{k}}M_{n_{k}}K_{s_{n_{k}}M_{n_{k}}}$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{r-1}\left(\prod_{j=1}^{k-1}r_{n_{j}}^{s_{n_{j}}}\right)n^{(k)}D_{s_{n_{k}}M_{n_{k}}}.$

We will also frequently use the next estimation of the Fejér kernels:

###### Corollary 1.3.10

Let $n\in\mathbb{N}.$ Then

$n\left|K_{n}\right|\leq c\sum_{l=\left\langle
n\right\rangle}^{\left|n\right|}M_{l}\left|K_{M_{l}}\right|\leq
c\sum_{l=0}^{\left|n\right|}M_{l}\left|K_{M_{l}}\right|$ (1.3.11)

where $c$ is an absolute constant.

###### Lemma 1.3.11

(see [1] and [31]) Let $n\in\mathbb{N}.$ Then, for any $n,N\in\mathbb{N_{+}}$,
we have that

$\displaystyle\int_{G_{m}}K_{n}(x)d\mu(x)=1,$ (1.3.12)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|K_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (1.3.13) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|K_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (1.3.14)

where $c$ is an absolute constant.

###### Lemma 1.3.12

(see [4, 9, 10, 54]) Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
decreasing numbers, satisfying the condition (2.1.6). Then for any
$n,N\in\mathbb{N_{+}},$

$\displaystyle\int_{G_{m}}F^{-1}_{n}(x)d\mu(x)=1,$ (1.3.15)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F^{-1}_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (1.3.16) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F^{-1}_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (1.3.17)

where $c$ is an absolute constant.

##  Chapter 2 Nörlund Means of Vilenkin-Fourier series in Lebesgue Spaces

###  2.1 Introduction

In the literature, a point $x\in G_{m}$ is called a Lebesgue point of $f\in
L^{1}\left(G_{m}\right),$ if

$\lim_{n\rightarrow\infty}M_{n}\int_{I_{n}(x)}f\left(t\right)dt=f\left(x\right)\
\ \ \ \ a.e.\ x\in G_{m}.$

It is well-known (for details see [88] and [61]) that if $f\in
L^{1}\left(G_{m}\right)$ then almost every point is a Lebesgue point and the
following important result holds true:

###### Proposition 2.1.1

Let $f\in L^{1}(G_{m})$. Then, for all Lebesgue points $x$,

$\underset{n\rightarrow\infty}{\lim}S_{M_{n}}f(x)=f(x)$

In the literature there are notations of $n$-th Nörlund $L_{n}$ and Riesz
$R_{n}$ logarithmic means defined by

$\displaystyle L_{n}f$ $\displaystyle:=$
$\displaystyle\frac{1}{l_{n}}\sum_{k=0}^{n-1}\frac{S_{k}f}{n-k},$
$\displaystyle R_{n}f$ $\displaystyle:=$
$\displaystyle\frac{1}{l_{n}}\sum_{k=1}^{n}\frac{S_{k}f}{k},$

respectively, where

$l_{n}:=\sum_{k=1}^{n}\frac{1}{k}.$

It is known that the Nörlund logarithmic mean has better approximation
properties then the partial sums and that the Riesz logarithmic means is
better than Fejér means in the same sense, but they have much more similar
properties with them. In [23] Gát and Goginava proved some convergence and
divergence properties of the Nörlund logarithmic means of functions in the
class of continuous functions and in the Lebesgue space $L^{1}.$ Moreover, Gát
and Goginava [24] proved that for each measurable function satisfying

$\phi\left(u\right)=o\left(u\log^{1/2}u\right),\text{ \ as \
}u\rightarrow\infty,$

there exists an integrable function $f$ such that

$\int_{G_{m}}\phi\left(\left|f\left(x\right)\right|\right)d\mu\,\left(x\right)<\infty$

and that there exists a set with positive measure such that the Nörlund
logarithmic means of the function diverges on this set. It follows that
weak-(1,1) type inequality does not hold for the maximal operator of Nörlund
logarithmic means:

$\displaystyle L^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|L_{n}f\right|$

but there exists an absolute constant $C_{p}$ such that

$\left\|L^{\ast}f\right\|_{p}\leq C_{p}\left\|f\right\|_{p},\text{ \ when \
}f\in L^{p},\text{ \ }p>1.$

Moreover (for details see [3]), if we consider the following restricted
maximal operator $\widetilde{L}_{\\#}^{\ast}f,$ defined by

$\displaystyle\widetilde{L}_{\\#}^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|L_{M_{n}}f\right|,\ \ \
(M_{k}:=m_{0}...m_{k-1},\ \ \ k=0,1...),$

then

$\displaystyle y\mu\left\\{\widetilde{L}_{\\#}^{\ast}f>y\right\\}\leq
c\left\|f\right\|_{1},\ \ \ f\in L^{1}(G_{m}),\ \ y>0.$

Hence, if $f\in L^{1}(G_{m}),$ then

$L_{M_{n}}f\to f,\ \ \text{a.e. on }\ \ G_{m}.$

Fejér’s theorem shows that (see e.g books [15] and [16]) if one replaces
ordinary summation by Cesàro summation $\sigma_{n}$ defined by

$\sigma_{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f,$

then the Féjer means of Fourier series of any integrable function converges
a.e on $G_{m}$ to the function.

Goginava and Gogoladze [27] introduced the operator $W_{A}$ defined by

$\displaystyle
W_{A}f(x):=\sum_{s=0}^{A-1}M_{s}\sum_{r_{s}=1}^{m_{s}-1}\int_{I_{A}(x-r_{s}e_{s})}\left|f(t)-f\left(x\right)\right|d\mu(t).$

and define a Vilenkin-Lebesgue point of function $f\in L^{1}(G_{m}),$ as a
point for which

$\displaystyle\lim_{A\rightarrow\infty}W_{A}f(x)=0$

Moreover, they also proved that the following result is true:

###### Proposition 2.1.2

Let $f\in L^{1}(G_{m})$. Then

$\underset{n\rightarrow\infty}{\lim}\sigma_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f$.

Moreover, the following is true:

###### Proposition 2.1.3

Let $x\in G_{m}$ and $f\in L^{1}(G_{m})$ is continuous at the point $x$. Then

$\underset{m\to\infty}{\mathop{\lim}}\,{{\sigma}_{m}}f\left(x\right)=f\left(x\right).$

If we consider the maximal operator of Féjer means $\sigma^{\ast}$ defined by:

$\displaystyle\sigma^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|\sigma_{n}f\right|$

then

$\displaystyle y\mu\left\\{\sigma^{\ast}f>y\right\\}\leq
c\left\|f\right\|_{1},\ \ \ f\in L^{1}(G_{m}),\ \ y>0.$

This result can be found in Zygmund [102] for trigonometric series, in Schipp
[58] for Walsh series and in Pál, Simon [48] for bounded Vilenkin series.

The boundedness does not hold from Lebesgue space $L^{1}(G_{m})$ to the space
$L^{1}(G_{m})$. On the other hand, if we consider restricted maximal operator
$\widetilde{\sigma}_{\\#}^{\ast}$ of Féjer means defined by

$\displaystyle\widetilde{\sigma}_{\\#}^{\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|\sigma_{M_{n}}f\right|$

then there exists a function $f\in L^{1}(G_{m})$ such that

$\|\widetilde{\sigma}_{\\#}^{\ast}f\|_{1}=\infty.$

In the one-dimensional case Yano [97] proved that

$\left\|\sigma_{n}f-f\right\|_{p}\rightarrow 0,\text{ \ \ \ as \ \ \ \
}n\rightarrow\infty,\text{ \ }(f\in L^{p}(G_{m}),\text{ \ }1\leq
p\leq\infty).$

However (see [32, 61]) the rate of convergence can not be better then
$O\left(n^{-1}\right)$ $\left(n\rightarrow\infty\right)$ for non-constant
functions, i.e., if $f\in L^{p},$ $1\leq p\leq\infty$ and

$\left\|\sigma_{M_{n}}f-f\right\|_{p}=o\left(\frac{1}{M_{n}}\right),\text{ \
as \ \ }n\rightarrow\infty,$

then  $f$ is a constant function.

It is also known that (see e.g the books [1] and [61]) for any $1\leq
p\leq\infty$ and $n\in\mathbb{N}$ we have the following estimate

$\left\|\sigma_{n}f-f\right\|_{p}\leq
c_{p}\omega_{p}\left(\frac{1}{M_{N}},f\right)+c_{p}\sum_{s=0}^{N-1}\frac{M_{s}}{M_{N}}\omega_{p}\left(\frac{1}{M_{s}},f\right).$

where $\omega_{p}\left(\frac{1}{M_{n}},f\right)$ is the modulus of continuity
of function $f\in L^{p}:$

$\omega_{p}\left(\frac{1}{M_{n}},f\right):=\sup\limits_{h\in
I_{n}}\left\|f\left(\cdot-h\right)-f\left(\cdot\right)\right\|_{p}.$

By applying this estimate, we immediately obtain that if $f\in
lip\left(\alpha,p\right),$ i.e.,

$\omega_{p}\left(\frac{1}{M_{n}},f\right)=O\left(\frac{1}{M_{n}^{\alpha}}\right),\
\ n\rightarrow\infty,$

then

$\left\|\sigma_{n}f-f\right\|_{p}=\left\\{\begin{array}[]{ll}O\left(\frac{1}{M_{N}}\right),&\text{if}\
\alpha>1,\\\ O\left(\frac{N}{M_{N}}\right),&\text{if}\ \alpha=1,\\\
O\left(\frac{1}{M_{N}^{\alpha}}\right),&\text{if}\
\alpha<1.\end{array}\right.$

Another well-known summability method is the so called
$\left(C,\alpha\right)$-means (Cesàro means) $\sigma_{n}^{\alpha}$, which are
defined by

$\sigma_{n}^{\alpha}f:=\frac{1}{A_{n}^{\alpha}}\overset{n}{\underset{k=1}{\sum}}A_{n-k}^{\alpha-1}S_{k}f,$

where

$A_{0}^{\alpha}:=0,\qquad
A_{n}^{\alpha}:=\frac{\left(\alpha+1\right)...\left(\alpha+n\right)}{n!},\qquad\alpha\neq-1,-2,...$

It is well-known that for $\alpha=1$ this summability method coincides with
the Fejér summation and for $\alpha=0$ we just have the partial sums of the
Vilenkin-Fourier series. Moreover, if we consider the maximal operator of the
Cesáro means $\sigma^{\alpha,\ast},$ defined by:

$\displaystyle\sigma^{\alpha,\ast}f$ $\displaystyle:=$
$\displaystyle\sup_{n\in\mathbb{N}}\left|\sigma_{n}^{\alpha}f\right|$

for $0<\alpha\leq 1,$ then

$\displaystyle y\mu\left\\{\sigma^{\alpha,\ast}f>y\right\\}\leq
c\left\|f\right\|_{1},\ \ f\in L^{1}(G_{m}),\ \ y>0.$

The boundedness of the maximal operator of the Reisz logarithmic means does
not hold from $L^{1}(G_{m})$ to the space $L^{1}(G_{m}).$ However,

$\left\|\sigma_{n}^{\alpha}f-f\right\|_{p}\rightarrow 0,\text{ \ \ \ when \ \
\ \ }n\rightarrow\infty,\text{ \ }(f\in L^{p}(G_{m}),\text{ \ }1\leq
p\leq\infty).$

Convergence and approximation in various norms of Vilenkin-Fejér means,
$\left(C,\alpha\right)$-means and Nörlund logarithmic means can be found in
Blahota, Gát and Goginava [5, 6], Blahota, and Tephnadze [8, 9, 10, 11, 12,
13, 14], Fine [17, 18], Fridli [19], Fujii [20], Goginava [25, 26],
Gogolashvili Nagy and Tephnadze [28, 29, 30], Persson and Tephnadze [50, 51]
(see also [3, 2, 4, 33, 34, 35, 49, 53, 52, 54, 55, 56, 57]), Pál and Simon
[48], Schipp [58, 59, 60], Simon [62, 63], Tephnadze [86, 64, 65, 66, 67, 71]
(see also [68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85,
87], Tutberidze [89, 90], Weisz [94, 95, 96] and Zhizhiashvili [98, 99, 100].
Similar problems for the two-dimensional case can be found in Nagy [39, 40,
41, 42], Nagy and Tephnadze [44, 43, 45, 46, 47].

The properties established in Lemma 1.3.11 ensure that kernel of the Fejér
means $\\{K_{N}\\}_{N=1}^{\infty}$ form what is called an approximation
identity.

###### Definition 2.1.4

The family $\\{\Phi_{n}\\}^{\infty}_{n=1}\subset L^{\infty}(G_{m})$ forms an
approximate identity provided that

$\displaystyle(A1)$ $\displaystyle\ \ \ \int_{G_{m}}\Phi_{n}(x)d(x)=1$
$\displaystyle(A2)$ $\displaystyle\ \ \
\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|\Phi_{n}(x)\right|d\mu(x)<\infty$
$\displaystyle(A3)$ $\displaystyle\ \ \
\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|\Phi_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,\ \ \text{for any}\ \ N\in\mathbb{N_{+}}.$

The term ”approximate identity” is used because of the fact that

$\Phi_{n}\ast f\to f\ \ \ \text{ as }\ \ \ n\to\infty$

in any reasonable sense. In particular, the following results holds true (for
details see the books [21] and [38]):

###### Proposition 2.1.5

Let $f\in L^{p}(G_{m}),$ where $1\leq p\leq\infty$ and the family

$\\{\Phi_{n}\\}^{\infty}_{n=1}\subset L^{\infty}(G_{m})$

forms an approximate identity. Then

$\|\Phi_{n}\ast f-f\|_{p}\to 0\ \ \text{ as }\ \ n\to\infty.$

It is well-known that the $n$-th Nörlund mean $t_{n}$ and $T$ means $T_{n}$
for the Fourier series of $f$ are, respectively, defined by

$t_{n}f:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}q_{n-k}S_{k}f$
(2.1.1)

and

$T_{n}f:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}S_{k}f,$
(2.1.2)

where $\\{q_{k}:k\in\mathbb{N}\\}$ is a sequence of nonnegative numbers and

$Q_{n}:=\sum_{k=0}^{n-1}q_{k}.$

Let $\\{q_{k}:k\geq 0\\}$ be a sequence of nonnegative numbers where
$q_{0}>0.$ Then the summability method (2.1.2) generated by $\\{q_{k}:k\geq
0\\}$ is regular if and only if (see [36])

$\lim_{n\rightarrow\infty}\frac{q_{n-1}}{Q_{n}}=\infty.$

The representations

$t_{n}f\left(x\right)=\underset{G_{m}}{\int}f\left(t\right)F_{n}\left(x-t\right)d\mu\left(t\right)$

and

$T_{n}f\left(x\right)=\underset{G_{m}}{\int}f\left(t\right)F^{-1}_{n}\left(x-t\right)d\mu\left(t\right)$

play central roles in the sequel, where

$F_{n}:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}q_{n-k}D_{k}$ (2.1.3)

and

$F^{-1}_{n}:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}q_{k}D_{k}$
(2.1.4)

are called the kernels of Nörlund and $T$ means, respectively.

Nörlund are generalizations of Fejér, $(C,\alpha)$ and Nörlund logarithmic
means. According to all these facts it is of prior interest to study the
behavior of operators related to Nörlund means of Fourier series with respect
to orthonormal systems. The Nörlund summation are general summability methods,
which satisfy the conditions (A1)-(A3). This means that all Nörlund and $T$
means are approximation identity. According to all these facts it is of prior
interest to study the behavior of operators related to Nörlund means of
Fourier series with respect to orthonormal systems. Móricz and Siddiqi [37]
investigated the approximation properties of some special Nörlund means of
Walsh-Fourier series of $L^{p}$ functions in norm. In particular, they proved
that if $f\in L^{p}(G_{m}),$ $1\leq p\leq\infty,$ $n=M_{j}+k,$ $1\leq k\leq
M_{j}\ (n\in\mathbb{N}_{+})$ and $\\{q_{k}:k\in\mathbb{N}_{+}\\}$ is sequence
of non-negative numbers, such that

$\frac{n^{\alpha-1}}{Q_{n}^{\alpha}}\sum_{k=0}^{n-1}q^{\alpha}_{k}=O(1),\ \
\text{for some}\ \ 1<\alpha\leq 2,$

then

$\|t_{n}f-f\|_{p}\leq\frac{C_{p}}{Q_{n}}\sum_{i=0}^{n-1}M_{i}q_{n-M_{i}}\omega_{p}\left(\frac{1}{M_{i}},f\right)+C_{p}\omega_{p}\left(\frac{1}{M_{j}},f\right),$

when $\\{q_{k}:k\in\mathbb{N}\\}$ is non-decreasing, while

$\displaystyle\|t_{n}f-f\|_{p}$ $\displaystyle\leq$
$\displaystyle\frac{C_{p}}{Q_{n}}\sum_{i=0}^{n-1}\left(Q_{n-M_{j}+1}-Q_{n-M_{j+1}+1}\right)\omega_{p}\left(\frac{1}{M_{i}},f\right)+C_{p}\omega_{p}\left(\frac{1}{M_{j}},f\right),$

when $\\{q_{k}:k\in\mathbb{N}\\}$ is non-increasing.

Let us define maximal operator of Nörlund means by

$\displaystyle t^{\ast}f:=\sup_{n\in\mathbb{N}}\left|t_{n}f\right|.$

If $\\{q_{k}:k\in\mathbb{N}\\}$ is non-increasing and satisfying the condition

$\frac{1}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \
}n\rightarrow\infty,$ (2.1.5)

or if $\\{q_{k}:k\in\mathbb{N}\\}$ is non-decreasing, satisfying the condition

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \
}n\rightarrow\infty,$ (2.1.6)

then

$\displaystyle y\mu\left\\{t^{\ast}f>y\right\\}\leq c\left\|f\right\|_{1},\ \
\ f\in L^{1}(G_{m}),\ \ y>0.$

The boundedness of the such maximal operator of Nörlund means does not hold
from $L^{1}(G_{m})$ to the space $L^{1}(G_{m}).$ However

$\left\|t_{n}f-f\right\|_{p}\rightarrow 0,\text{ \ \ \ as \ \ \ \
}n\rightarrow\infty,\text{ \ }(f\in L^{p}(G_{m}),\text{ \ }1\leq
p\leq\infty).$

###  2.2 Well-known and New examples of Nörlund Means

We define $B_{n}$ means as the class of Nörlund means, with monotone and
bounded sequence $\\{q_{k}:k\in\mathbb{N}\\}$, such that

$0<q<\infty\ \ \ \text{where}\ \ \
q_{\infty}:=\lim_{n\rightarrow\infty}q_{n}.$

If the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ is non-decreasing, then we have
that

$nq_{0}\leq Q_{n}\leq nq_{\infty}.$

In the case when the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ is non-increasing,
then

$nq_{\infty}\leq Q_{n}\leq nq_{0}.$

In both cases we can conclude that conditions (2.1.6) and (2.1.5) are
fulfilled.

Well-known examples of Nörlund means with monotone and bounded sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ is Fejér means

$\sigma_{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f$

It is evident that in this case conditions (2.1.6) and (2.1.5) are fulfilled.

The Cesàro means $\sigma_{n}^{\alpha}$ (sometimes also denoted
$\left(C,\alpha\right)$) are defined by

$\sigma_{n}^{\alpha}f:=\frac{1}{A_{n}^{\alpha}}\overset{n}{\underset{k=1}{\sum}}A_{n-k}^{\alpha-1}S_{k}f$

where

$A_{0}^{\alpha}:=0,\qquad
A_{n}^{\alpha}:=\frac{\left(\alpha+1\right)...\left(\alpha+n\right)}{n!},\qquad\alpha\neq-1,-2,...$

It is well-known that

$A_{n}^{\alpha}=\overset{n}{\underset{k=0}{\sum}}A_{n-k}^{\alpha-1},$ (2.2.1)
$A_{n}^{\alpha}-A_{n-1}^{\alpha}=A_{n}^{\alpha-1}\ \ \ \text{and}\ \ \
A_{n}^{\alpha}\backsim n^{\alpha}.$ (2.2.2)

It is obvious that

$\frac{\left|q_{n}-q_{n+1}\right|}{n^{\alpha-2}}=O\left(1\right),\qquad\text{as}\qquad
n\rightarrow\infty,$ (2.2.3)
$\frac{q_{0}}{Q_{n}}=O\left(\frac{1}{n^{\alpha}}\right),\qquad\ \ \ \ \
\text{as}\qquad n\rightarrow\infty,$ (2.2.4)

and

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\qquad\ \ \ \ \
\text{as}\qquad n\rightarrow\infty.$ (2.2.5)

Let $V_{n}^{\alpha}$ denote the Nörlund mean, where

$\left\\{q_{k}=\left(k+1\right)^{\alpha-1}:\quad k\in\mathbb{N},\quad
0<\alpha<1\right\\},$

that is

$V_{n}^{\alpha}f:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}\left(n-k-1\right)^{\alpha-1}S_{k}f.$

It is obvious that

$\frac{\left|q_{n}-q_{n+1}\right|}{n^{\alpha-2}}=O\left(1\right),\qquad\text{as}\qquad
n\rightarrow\infty,$ (2.2.6)
$\frac{q_{0}}{Q_{n}}=O\left(\frac{1}{n^{\alpha}}\right),\qquad\ \ \ \ \
\text{as}\qquad n\rightarrow\infty,$ (2.2.7)

and

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\qquad\ \ \ \ \
\text{as}\qquad n\rightarrow\infty.$ (2.2.8)

We just remind again that $n$-th Nörlund $L_{n}$ and Riesz $R_{n}$ logarithmic
means are defined by the sequence $\left\\{q_{k}=1/k,\quad
k\in\mathbb{N}_{+}\right\\}:$

$\displaystyle L_{n}f:=\frac{1}{l_{n}}\sum_{k=0}^{n-1}\frac{S_{k}f}{n-k},\ \ \
R_{n}f:=\frac{1}{l_{n}}\sum_{k=1}^{n}\frac{S_{k}f}{k},$

respectively, where

$l_{n}:=\sum_{k=1}^{n}\frac{1}{k}.$

It is evident that

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{ \ \ as \ \
}n\rightarrow\infty$ (2.2.9)

and

$\frac{q_{0}}{Q_{n}}=O\left(\frac{1}{\ln n}\right),\text{ \ as \
}n\rightarrow\infty.$ (2.2.10)

Let $U^{\alpha}_{n}$ denote the Nörlund mean, where

$\left\\{q_{k}=\frac{1}{(k+3){\ln^{\alpha}(k+3)}}\ :\ k\in\mathbb{N},\
0<\alpha\leq 1\right\\},$

that is

$U_{n}^{\alpha}f:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}\frac{S_{k}f}{\left(n-k-3\right)\ln^{\alpha}\left(n-k-3\right)}.$

It is obvious that

$\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\qquad\text{as}\qquad
n\rightarrow\infty$ (2.2.11)

and

$\frac{q_{0}}{Q_{n}}=\left\\{\begin{array}[]{c}O\left(1/\ln(\ln
n\right)),\text{ \ \ \ if \ \ \ }\alpha=1,\\\
O\left(1/\ln^{1-\alpha}n\right),\text{ \ \ \ \ \ if
}0<\alpha<1.\end{array}\right.$ (2.2.12)

Let $\alpha\in\mathbb{R}_{+}$. If we define the sequence
$\\{q_{k}:k\in\mathbb{N}\\}$ by

$\left\\{q_{k}=\log^{\alpha}(k+1)\ :\ k\in\mathbb{N}\ :\ \alpha>0\right\\},$

then we get the class of Nörlund means with non-decreasing coefficients:

$\beta_{n}^{\alpha}f:=\frac{1}{Q_{n}}\sum_{k=1}^{n}\log^{\alpha}\left(n-k-1\right)S_{k}f.$

It is obvious that

$\frac{n}{2}\log^{\alpha}\left(n/2\right)\leq Q_{n}\leq n\log^{\alpha}n.$

It follows that

$\displaystyle\frac{1}{Q_{n}}\leq\frac{c}{n\log^{\alpha}n}=O\left(\frac{1}{n}\right)\rightarrow
0,\text{ \ as \ }n\rightarrow\infty.$ (2.2.13)

and

$\displaystyle\frac{q_{n-1}}{Q_{n}}\leq\frac{c\log^{\alpha}\left(n-1\right)}{n\log^{\alpha}n}=O\left(\frac{1}{n}\right)\rightarrow
0,\text{ \ as \ }n\rightarrow\infty.$ (2.2.14)

###  2.3 Kernels of Nörlund Means

Now we study kernels of Nörlund means with respect to Vilenkin systems. If we
invoke Abel transformations for $a_{j}=A_{j}-A_{j-1},\ j=1,...,n,$

$\displaystyle\overset{n}{\underset{j=1}{\sum}}a_{j}b_{n-j}$ $\displaystyle=$
$\displaystyle
A_{n}b_{0}+\overset{n-1}{\underset{j=1}{\sum}}A_{j}(b_{j}-b_{j+1}),$ (2.3.1)
$\displaystyle\overset{n}{\underset{j=M_{N}}{\sum}}a_{j}b_{n-j}$
$\displaystyle=$ $\displaystyle
A_{n}b_{0}-A_{M_{N}-1}b_{n-M_{N}}+\overset{n-1}{\underset{j=M_{N}}{\sum}}A_{j}(b_{j}-b_{j+1}),$
(2.3.2)

when $b_{j}=q_{j}$, $a_{j}=1$ and $A_{j}=j$ for $j=0,1,...,n,$ then (2.3.1)
and (2.3.2) give the following identities:

$\displaystyle Q_{n}$ $\displaystyle:=$
$\displaystyle\overset{n-1}{\underset{j=0}{\sum}}q_{j}=\overset{n}{\underset{j=1}{\sum}}q_{n-j}\cdot
1=\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j+q_{0}n,$
(2.3.3) $\displaystyle=$
$\displaystyle\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{n-j}=\overset{n-1}{\underset{j=M_{N}}{\sum}}q_{n-j}\cdot
1$ $\displaystyle=$
$\displaystyle\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j+q_{0}n-(M_{N}-1)q_{n-M_{N}}.$

Moreover, if we instead use the Abel transformations (2.3.1) and (2.3.2) for
$b_{j}=q_{n-j}$, $a_{j}=D_{j}$ and $A_{j}=jK_{j}$ for any $j=0,1,...,n-1$ we
get the identities:

$\displaystyle F_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)jK_{j}+q_{0}nK_{n}\right),$
(2.3.5) $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{n-j}D_{j}$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)jK_{j}+q_{0}nK_{n}-q_{n-M_{N}}(M_{N}-1)K_{M_{N}-1}\right).$

Analogously, if we use the Abel transformations (2.3.1) and (2.3.2) for
$b_{j}=q_{j}$, $a_{j}=S_{j}$ and $A_{j}=j\sigma_{j}$ for any $j=0,1,...,n-1$
we get the identities:

$\displaystyle t_{n}f$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\sigma_{j}f+q_{0}n\sigma_{n}f\right)$
$\displaystyle\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{n-j}S_{j}f$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\sigma_{j}f+q_{0}nK_{n}-q_{n-M_{N}}(M_{N}-1)\sigma_{M_{N}-1}f\right).$

First we consider Nörlund kernels with respect to Vilenkin systems, which are
generated by non-decreasing sequences:

###### Lemma 2.3.1

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers,
satisfying the condition (2.1.6). Then

$\left|F_{n}\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\},$

where $c$ is an absolute constant.

  * Proof. Let the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-decreasing. Then, by using (2.1.6), we get that

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|+q_{0}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)+q_{0}\right)$
$\displaystyle\leq$ $\displaystyle\frac{q_{n-1}}{Q_{n}}\leq\frac{c}{n}.$

Hence, in view of (2.1.6) if we apply (1.3.11) in Corollary 1.3.10 and use the
equality (2.3.5) we obtain that

$\displaystyle\left|F_{n}\right|$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle=$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{q_{n-1}}{Q_{n}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|\leq\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete.

We also state analogical estimate, but now without any restriction like
(2.1.6):

###### Lemma 2.3.2

Let $n\geq M_{N}$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
decreasing numbers. Then

$\left|\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{n-j}D_{j}\right|\leq\frac{c}{M_{N}}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\},$

where $c$ is an absolute constant.

  * Proof. Let $M_{N}-1\leq j\leq n.$ In the view of (1.3.11) in Corollary 1.3.10 we find that

$\displaystyle\left|K_{j}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{j}\sum_{l=0}^{\left|j\right|}M_{l}\left|K_{M_{l}}\right|\leq\frac{1}{M_{N}-1}\sum_{l=0}^{\left|n\right|}M_{l}\left|K_{M_{l}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{c}{M_{N}}\sum_{l=0}^{\left|n\right|}M_{l}\left|K_{M_{l}}\right|$

Since the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-decreasing we get that

$\displaystyle q_{n-M_{N}}(M_{N}-1)$ $\displaystyle=$
$\displaystyle\overset{M_{N}-1}{\underset{j=0}{\sum}}q_{n-M_{N}}\leq\overset{M_{N}-1}{\underset{j=0}{\sum}}q_{n-M_{N}+j}$
$\displaystyle=$ $\displaystyle
q_{n-M_{N}}+q_{n-M_{N}+2}+...+q_{n-M_{N}+(M_{N}-1)}\leq Q_{n}$

and

$\displaystyle\overset{n-1}{\underset{j=M_{N}}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|j+q_{0}n+q_{n-M_{N}}(M_{N}-1)$
$\displaystyle\leq$
$\displaystyle\overset{n-1}{\underset{j=1}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|j+q_{0}n+q_{n-M_{N}}(M_{N}-1)$
$\displaystyle=$
$\displaystyle\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j+q_{0}n+q_{n-M_{N}}(M_{N}-1)$
$\displaystyle=$ $\displaystyle Q_{n}+q_{n-M_{N}}(M_{N}-1)\leq 2Q_{n}.$

By using the Abel transformation (2.3) we can conclude that

$\displaystyle\left|\frac{1}{Q_{n}}\overset{n}{\underset{j=M_{N}}{\sum}}q_{n-j}D_{j}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)jK_{j}+q_{0}nK_{n}+q_{n-M_{N}}(M_{N}-1)K_{M_{N}-1}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=M_{N}}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|j+q_{0}n\right)\right)\frac{c}{M_{N}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{c}{M_{N}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete.

Now we prove a lemma, which is very important for our further investigation to
prove norm convergence in Lebesgue spaces of Nörlund means generated by non-
decreasing sequences $\\{q_{k}:k\in\mathbb{N}\\}$ in this chapter.

###### Lemma 2.3.3

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers.
Then, for any $n,N\in\mathbb{N_{+}}$,

$\displaystyle\int_{G_{m}}F_{n}(x)d\mu(x)=1,$ (2.3.9)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (2.3.10) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (2.3.11)

where $c$ is an absolute constant.

  * Proof. According to Lemma 1.3.4 we readily obtain (2.3.9). By using (1.3.13) in Corollary 1.3.11 combined with (2.3.3) and (2.3.5) we get that

$\displaystyle\int_{G_{m}}\left|F_{n}\right|d\mu$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\int_{G_{m}}\left|K_{j}\right|d\mu+\frac{q_{0}n}{Q_{n}}\int_{G_{m}}|K_{n}|d\mu$
$\displaystyle\leq$
$\displaystyle\frac{c}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j+\frac{cq_{0}n}{Q_{n}}<c<\infty,$

so also (2.3.10) is proved.

According to (1.3.14) in Corollary 1.3.11 and also (2.3.3) and (2.3.5) we find
that

$\displaystyle\int_{G_{m}\backslash I_{N}}\left|F_{n}\right|d\mu$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\int_{G_{m}\backslash
I_{N}}\left|K_{j}\right|d\mu$ $\displaystyle+$
$\displaystyle\frac{q_{0}n}{Q_{n}}\int_{G_{m}\backslash I_{N}}|K_{n}|$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}+\frac{q_{0}n\alpha_{n}}{Q_{n}},$
$\displaystyle:=$ $\displaystyle I+II.$

where $\alpha_{n}\to 0,\ \ \text{as}\ \ n\to\infty.$ Since the sequence is
non-decreasing, we can conclude that

$II=\frac{q_{0}n\alpha_{n}}{Q_{n}}\leq\alpha_{n}\to 0,\ \ \text{as}\ \
n\to\infty.$

On the other hand, for any $\varepsilon>0$ there exists $N_{0}\in\mathbb{N},$
such that

$\alpha_{n}<\varepsilon,\ \ \text{ when }\ \ n>N_{0}.$

Moreover,

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}+\frac{1}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}=I_{1}+I_{2}.$

Since sequence is non-decreasing, we can conclude that

$|q_{n-j}-q_{n-j-1}|<2q_{n-1}$ $\displaystyle
I_{1}=\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}\leq\frac{2q_{n-1}N_{0}}{Q_{n}}\to
0,\ \ \ \text{as}\ \ \ n\to\infty$

and, furthermore,

$\displaystyle
I_{2}=\frac{1}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j<\varepsilon,$

and it follows that also $I\to 0$ so also (2.3.11) is proved. The proof is
complete.

We also consider the kernel of Nörlund means with respect to the Vilenkin
systems which are generated by non-increasing sequences
$\\{q_{k}:k\in\mathbb{N}\\}$, but now with some new restrictions on the
indexes:

###### Lemma 2.3.4

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers
satisfying the condition (2.1.5). Then

$\left|F_{n}\right|\leq\frac{c}{n}\left\\{\sum_{j=0}^{\left|n\right|}M_{j}\left|K_{M_{j}}\right|\right\\},$

where $c$ is an absolute constant.

  * Proof. Let the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ be non-increasing and satisfying condition (2.1.5). Then

$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|+q_{0}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}-\left(q_{n-j}-q_{n-j-1}\right)+q_{0}\right)$
$\displaystyle\leq$
$\displaystyle\frac{2q_{0}-q_{n-1}}{Q_{n}}\leq\frac{2q_{0}}{Q_{n}}\leq\frac{c}{n}.$

Hence, if we apply (1.3.11) in Corollary 1.3.10 and invoke equalities (2.3.3)
and (2.3.5), then we get that

$\displaystyle\left|F_{n}\right|$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left|q_{n-j}-q_{n-j-1}\right|+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle=$
$\displaystyle\left(\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}-\left(q_{n-j}-q_{n-j-1}\right)+q_{0}\right)\right)\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{2q_{0}-q_{n-1}}{Q_{n}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{2q_{0}}{Q_{n}}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{c}{n}\sum_{i=0}^{\left|n\right|}M_{i}\left|K_{M_{i}}\right|.$

The proof is complete.

The next result is very important for our further investigation in this
Chapter to prove norm convergence in Lebesgue spaces of Nörlund means
generated by a non-increasing sequence $\\{q_{k}:k\in\mathbb{N}\\}$:

###### Corollary 2.3.5

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers
satisfying the condition (2.1.5). Then, for any $n,N\in\mathbb{N_{+}}$,

$\displaystyle\int_{G_{m}}F_{n}(x)d\mu(x)=1,$ (2.3.12)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F_{n}(x)\right|d\mu(x)\leq
c<\infty,$ (2.3.13) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F_{n}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (2.3.14)

where $c$ is an absolute constant.

  * Proof. If we compare the estimation of $K_{n}$ in Lemma 1.3.10 or $F_{n}$ in Lemma 2.3.1 with the estimation of $F_{n}$ in Lemma 2.3.4 we find that they are quite the same. Hence, the proof is analogous to those of Corollary 1.3.11 and Lemma 2.3.3, so, we leave out the details.

Finally we study some special subsequences of kernels of Nörlund and $T$
means:

###### Lemma 2.3.6

Let $n\in\mathbb{N}$. Then

$\displaystyle
F_{M_{n}}(x)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{F^{-1}}_{M_{n}}(x)$
(2.3.15)

and

$\displaystyle
F^{-1}_{M_{n}}(x)=D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{F}_{M_{n}}(x).$
(2.3.16)

  * Proof. By using (1.3.1) in Lemma 1.3.1 we get that

$\displaystyle F_{M_{n}}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}}{\underset{k=1}{\sum}}q_{M_{n}-k}D_{k}(x)=\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}q_{k}D_{M_{n}-k}(x)$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}q_{k}\left(D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{D}_{j}(x)\right)$
$\displaystyle=$ $\displaystyle
D_{M_{n}}(x)-\psi_{M_{n}-1}(x)\overline{F^{-1}}_{M_{n}}(x)$

Hence, (2.3.15) is proved. Identity (2.3.16) is proved analogously so the
proof is complete.

Next four lemmas will be used to prove norm convergence and almost everywhere
convergence of subsequences of Nörlund means:

###### Corollary 2.3.7

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-decreasing numbers.
Then, for any $n,N\in\mathbb{N_{+}}$,

$\displaystyle\int_{G_{m}}F^{-1}_{M_{n}}(x)d\mu(x)=1,$ (2.3.17)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F^{-1}_{M_{n}}(x)\right|d\mu(x)\leq
c<\infty,$ (2.3.18) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F^{-1}_{M_{n}}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (2.3.19)

  * Proof. According to (2.3.16) the proof is a direct consequence of Lemmas 1.3.3 and 2.3.3 and Lemma 1.3.4. The proof is complete.

###### Corollary 2.3.8

Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing numbers.
Then, for any $N\in\mathbb{N_{+}},$

$\displaystyle\int_{G_{m}}F_{M_{n}}(x)d\mu(x)=1,$ (2.3.20)
$\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|F_{M_{n}}(x)\right|d\mu(x)\leq
c<\infty,$ (2.3.21) $\displaystyle\sup_{n\in\mathbb{N}}\int_{G_{m}\backslash
I_{N}}\left|F_{M_{n}}(x)\right|d\mu(x)\rightarrow 0,\ \ \text{as}\ \
n\rightarrow\infty,$ (2.3.22)

  * Proof. According to (2.3.15) the proof is a direct consequence of Lemmas 1.3.3, 1.3.4 and 1.3.12. The proof is complete.

###  2.4 Norm Convergence of Nörlund Means in Lebesgue Spaces

First we consider norm convergence of Nörlund means with respect to Vilenkin
systems:

###### Theorem 2.4.1

Let $f\in L^{p}(G_{m})$ for $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a
sequence of non-decreasing numbers. Then

$\|t_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty.$

  * Proof. According to Lemma 2.3.3 we conclude that the conditions (A1), (A2) and (A3) in Theorem 2.1.5 are fulfilled, which implies the stated norm convergence.

The proof is complete.

###### Theorem 2.4.2

Let $f\in L^{p}(G_{m})$ for $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a
sequence of non-increasing numbers satisfying the condition (2.1.5). Then

$\|t_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty.$

  * Proof. According to Corollary 2.3.5 we conclude that the conditions (A1), (A2) and (A3) in Theorem 2.1.5 are fulfilled and the stated norm convergence follows.

The proof is complete.

According to Theorems 2.4.1 and 2.4.2 we get the following result for Nörlund
means:

###### Corollary 2.4.3

Let $f\in L^{p}(G_{m})$ for $p\geq 1$. Then

$\displaystyle\|\sigma_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty,$
$\displaystyle\|B_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty,$
$\displaystyle\|\beta^{\alpha}_{n}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty,$

  * Proof. Since $\sigma_{n}f$ and $\beta^{\alpha}_{n}f$ are Nörlund means generated by non-decreasing sequences $\\{q_{k}:k\in\mathbb{N}\\},$ the corresponding norm convergences are direct consequences of Theorem 2.4.1. In the case of $B_{n}f$ means with non-decreasing sequence $\\{q_{k}:k\in\mathbb{N}\\},$ this result is also a consequence of Theorem 2.4.1.

On the other hand, in the case of $B_{n}f$ means with non-increasing sequence
$\\{q_{k}:k\in\mathbb{N}\\},$ this result is a consequence of Theorem 2.4.2
and (2.1.5).

The proof is complete.

Now, we consider subsequences of Nörlund means, generated by non-increasing
sequences, but without any restrictions on the sequence
$\\{q_{k}:k\in\mathbb{N}\\}$:

###### Theorem 2.4.4

Let $f\in L^{p}(G_{m})$ for $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a
sequence of non-increasing numbers. Then

$\|t_{M_{n}}f-f\|_{p}\to 0\ \ \text{as}\ \ n\to\infty.$

  * Proof. According to Corollary 2.3.8 we conclude that conditions (A1), (A2) and (A3) in Theorem 2.1.5 are fulfilled and the claimed norm convergence is proved. The proof is complete.

###  2.5 Convergence of Nörlund Means in Vilenkin-Lebesgue points

Our first main result concerning convergence of Nörlund means reads:

###### Theorem 2.5.1

a) Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
decreasing numbers.

If the function $f\in L^{1}(G_{m})$ is continuous at a point $x,$ then

${{t}_{n}}f(x)\to f(x),\ \ \ \text{as}\ \ \ n\to\infty.$

Furthermore,

$\underset{n\rightarrow\infty}{\lim}t_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f\in L^{p}(G_{m})$.

b) Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-
increasing numbers satisfying the condition (2.1.5).

If the function $f\in L^{1}(G_{m})$ is continuous at a point $x,$ then

${{t}_{n}}f(x)\to f(x),\ \ \ \text{as}\ \ \ n\to\infty.$

Moreover,

$\underset{n\rightarrow\infty}{\lim}t_{n}f(x)=f(x)$

for all Vilenkin-Lebesgue points of $f\in L^{p}(G_{m})$.

  * Proof. Let $\\{q_{k}:k\in\mathbb{N}\\}$ be a non-decreasing sequence. Suppose that $x$ is either a point of continuity of a function $f\in L^{p}(G_{m})$ or Vilenkin-Lebesgue point of the function $f\in L^{p}(G_{m}).$ According to Proposition 2.1.3 and Proposition 2.1.2 we can conclude that

$\underset{n\rightarrow\infty}{\lim}|\sigma_{n}f(x)-f(x)|=0.$

Hence, by combining (2.3.3) and (2.3) we can conclude that

$\displaystyle|t_{n}f(x)-f(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j|\sigma_{j}f(x)-f(x)|+q_{0}n|\sigma_{n}f(x)-f(x)|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}+\frac{q_{0}n\alpha_{n}}{Q_{n}}:=I+II,$

where $\alpha_{n}\to 0,\ \text{as}\ n\to\infty.$

Since the sequence $\\{q_{k}:k\in\mathbb{N}\\}$ is non-decreasing, we can
conclude that

$II\leq\alpha_{n}\to 0,\ \ \text{as}\ \ n\to\infty.$

On the other hand, since $\alpha_{n}$ converges to $0,$ we get that there
exists an absolute constant $A,$ such that $\alpha_{n}\leq A$ for any
$n\in\mathbb{N}$ and for any $\varepsilon>0$ there exists
$N_{0}\in\mathbb{N},$ such that

$\alpha_{n}<\varepsilon\ \ \text{when }\ \ n>N_{0}.$

Hence,

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}+\frac{1}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}$
$\displaystyle:=$ $\displaystyle I_{1}+I_{2}.$

Since

$|q_{n-j}-q_{n-j-1}|<2q_{n-1}\ \ \ \text{ and }\ \ \ \alpha_{n}<A,$

we obtain that

$\displaystyle
I_{1}=\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}\leq\frac{2AN_{0}q_{n-1}}{Q_{n}}\to
0,\ \ \ \text{as}\ \ \ n\to\infty$

and

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j\alpha_{j}$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{Q_{n}}\overset{n-1}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{Q_{n}}\overset{n-1}{\underset{j=0}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)j<\varepsilon.$

We conclude that also $I_{2}\to 0$ so a) is proved.

Assume now that the sequence is non-increasing and satisfying condition
(2.1.5). To prove convergence in Vilenkin-Lebesgue points we use the
estimations (2.3.3) and (2.3) to obtain that

$\displaystyle|t_{n}f-f(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=0}{\sum}}\left(q_{n-j-1}-q_{n-j}\right)j\alpha_{j}+\frac{q_{0}n\alpha_{n}}{Q_{n}}$
$\displaystyle:=$ $\displaystyle III+IV,$

where $\alpha_{n}\to 0,\ \text{as}\ n\to\infty.$

It is evident that

$IV\leq\frac{q_{0}n\alpha_{n}}{Q_{n}}\leq C\alpha_{n}\to 0,\ \ \text{as}\ \
n\to\infty.$

Moreover, for any $\varepsilon>0$ there exists $N_{0}\in\mathbb{N},$ such that
$\alpha_{n}<\varepsilon$ when $n>N_{0}.$ It follows that

$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=1}{\sum}}\left(q_{n-j-1}-q_{n-j}\right)j\alpha_{j}$
$\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{N_{0}}{\underset{j=1}{\sum}}\left(q_{n-j-1}-q_{n-j}\right)j\alpha_{j}+\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j-1}-q_{n-j}\right)j\alpha_{j}$
$\displaystyle:=$ $\displaystyle III_{1}+III_{2}.$

Since sequence is non-increasing, we can conclude that

$|q_{n-j}-q_{n-j-1}|<2q_{0}.$

Hence,

$\displaystyle III_{1}\leq\frac{2q_{0}N_{0}}{Q_{n}}\to 0,\ \ \ \text{as}\ \ \
n\to\infty$

and

$\displaystyle III_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j-1}-q_{n-j}\right)j\alpha_{j}$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon(n-1)}{Q_{n}}\overset{n-2}{\underset{j=N_{0}+1}{\sum}}\left(q_{n-j}-q_{n-j-1}\right)$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon(n-1)}{Q_{n}}\left(q_{0}-q_{n-N_{0}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{2q_{0}\varepsilon(n-1)}{Q_{n}}<C\varepsilon.$

Hence, also $III\to 0$ so the proof of part b) is also complete.

###### Corollary 2.5.2

Let $f\in L^{p}(G_{m}),$ where $p\geq 1.$ Then, for all Lebesgue points of
$f\in L^{p}(G_{m}),$

$\displaystyle\sigma_{n}f$ $\displaystyle\rightarrow$ $\displaystyle f,\text{
\ as \ }n\rightarrow\infty,$ $\displaystyle B_{n}f$ $\displaystyle\rightarrow$
$\displaystyle f,\text{ \ as \ }n\rightarrow\infty,$
$\displaystyle\beta^{\alpha}_{n}f$ $\displaystyle\rightarrow$ $\displaystyle
f,\text{ \ as \ }n\rightarrow\infty.$

###### Theorem 2.5.3

Let $p\geq 1$ and $\\{q_{k}:k\in\mathbb{N}\\}$ be a sequence of non-increasing
numbers. Then

$\underset{n\rightarrow\infty}{\lim}t_{M_{n}}f(x)=f(x)$

for all Lebesgue points of $f\in L^{p}(G_{m})$.

  * Proof. By using Lemma 2.3.6 we get that

$\displaystyle t_{M_{n}}f\left(x\right)$ $\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)F_{n}\left(x-t\right)d\mu\left(t\right)$
$\displaystyle=$
$\displaystyle\underset{G_{m}}{\int}f\left(t\right)D_{M_{n}}\left(x-t\right)d\mu\left(t\right)-\underset{G_{m}}{\int}f\left(t\right)\psi_{M_{n}-1}(x-t)\overline{F^{-1}}_{M_{n}}(x-t)$
$\displaystyle:=$ $\displaystyle I-II.$

By applying Proposition 2.1.1 we get that $I=S_{M_{n}}f(x)\to f(x)$ for all
Lebesgue points of $f\in L^{p}(G_{m})$, where $p\geq 1.$

Moreover, according to Proposition 1.1.1 we find that

$\psi_{M_{n}-1}(x-t)=\psi_{M_{n}-1}(x)\overline{\psi}_{M_{n}-1}(t)$

and

$II=\psi_{M_{n}-1}(x)\underset{G_{m}}{\int}f\left(t\right)\overline{F^{-1}}_{M_{n}}(x-t)\overline{\psi}_{M_{n}-1}(t)d(t).$

By combining Theorem 1.2.2 and Corollary 1.3.12 we find that the function

$f\left(t\right)\overline{F^{-1}}_{M_{n}}(x-t)\in L^{p}(G_{m})\ \ \text{
where}\ \ p\geq 1\ \ \text{for any }\ \ x\in G_{m},$

and $II$ are Fourier coefficients of an integrable function. Hence, according
to the Riemann-Lebesgue Lemma we get that

$II\to 0,\text{ \ as \ }n\rightarrow\infty,\ \text{for any }\ x\in G_{m}.$

The proof is complete.

###### Corollary 2.5.4

Let $f\in L^{p}(G_{m}),$ where $p\geq 1.$ Then, for all Lebesgue points of
$f\in L^{p}(G_{m}),$

$\displaystyle\sigma^{\alpha}_{M_{n}}f$ $\displaystyle\rightarrow$
$\displaystyle f,\text{ \ as \ }n\rightarrow\infty,$ $\displaystyle
V^{\alpha}_{M_{n}}f$ $\displaystyle\rightarrow$ $\displaystyle f,\text{ \ as \
}n\rightarrow\infty,$ $\displaystyle U^{\alpha}_{M_{n}}f$
$\displaystyle\rightarrow$ $\displaystyle f,\text{ \ as \
}n\rightarrow\infty.$

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# Siquieros accidental painting technique: a fluid mechanics point of view

Sandra Zetina1 and Roberto Zenit2
1. Instituto de Investigaciones Estéticas
2.Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México
Cd. Universitaria, México D.F., 04510
MÉXICO

###### Abstract

This is an entry for the Gallery of Fluid Motion of the 65st Annual Meeting of
the APS-DFD ( fluid dynamics video ). This video shows an analysis of the
‘accidental painting’ technique developed by D.A. Siqueiros, a famous Mexican
muralist. We reproduced the technique that he used: pouring layers of paint of
different colors on top of each other. We found that the layers mix, creating
aesthetically pleasing patterns, as a result of a Rayleigh-Taylor instability.
Due to the pigments used to give paints their color, they can have different
densities. When poured on top of each other, if the top layer is denser than
the lower one, the viscous gravity current undergoes unstable as it spread
radially. We photograph the process and produced slowed-down video to
visualize the process.

##  1 Introduction

In the spring of 1936, the famous Mexican muralist David Alfaro Siqueiros [1]
organized an experimental painting workshop in New York: a group of artists
focused in developing painting techniques through empirical experimentation of
modern and industrial materials and tools. Among the young artists attending
the workshop was Jackson Pollock [2]. They tested different lacquers and a
number of experimental techniques. One of the techniques, named by Siqueiros
as a “controlled accident,” consisted in pouring layers of paint of different
colors on top of each other. After a brief time, the paint from the lower
layer emerged from bottom to top creating a relatively regular pattern of
blobs. This technique led to the creation of explosion-inspired textures and
catastrophic images. We conducted an analysis of this process. We
experimentally reproduced the patterns “discovered” by Siqueiros and analyzed
the behavior of the flow. We found that the flow is driven by the well-known
Rayleigh Taylor instability [3]: different colors paints have different
densities; a heavy layer on top of a light one is an unstable configuration.
The blobs and plumes that result from the instability create the aesthetically
pleasing patterns. We discuss the importance of fluid mechanics in artistic
creation.

##  2 Experimental Conditions

We used the same type of paints that Siqueiros used. Their physical properties
are shown in Table 1. With these combinations, both the Reynolds and Artwood
numbers are small. The paint layers where dripping on top of a horizontal
glass sheet. Volumes of approximately 50 and 25 ml were deposited for the
bottom and bottom layers, respectively. The flow was photographed with a
computer-controlled HD digital camera (FinePix Si Pro, Fujifilm), such that
long term time sequences could be obtained. The time interval between photos
was either 500 or 143 ms.

Fluid | density, kg/m3 | viscosity Pa | Ar | Re
---|---|---|---|---
white paint | 1110 | 2.5 | 5.1$\times 10^{-2}$ | 6.9$\times 10^{-5}$
black paint | 1002 | 11.7 |  |
yellow paint | 1080 | 3.6 | 3.4$\times 10^{-2}$ | 1.1$\times 10^{-4}$
transparent lacquer | 1008 | 12.9 |  |
Table 1: Properties of the fluids and fluid combinations used.
$Ar=(\rho_{1}-\rho_{2})/(\rho_{1}+\rho_{2})$, is the Artwood number and
$Re=U_{f}H\rho/\mu$, is the Reynolds number, where $U_{f}$ and $H$ and the
velocity of the front and the thickness of the layer, respectively.

To our knowledge an formal analysis of this process has not been studied to
date.

##  3 Videos

Our video contributions can be found at:

  * •

Video 1, mpeg4, full resolution

  * •

Video 2, mpeg2, low resolution

## References

  * [1] P. Stein, _Siqueiros: His Life and Works_. (Intl. Pub., 1994).
  * [2] E. G. Landau, _Jackson Pollock_. (Abrams, 2010).
  * [3] F. Charru, _Hydrodynamic Instabilities_. (Cambridge University Press, 2011).