# On the divergence of Féjer means with respect to Vilenkin systems on the set
of measure zero

N. Areshidze, L.-E. Persson and G. Tephnadze  N. Areshidze, Department of
Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University,
Chavchavadze str. 1, Tbilisi 0128, Georgia<EMAIL_ADDRESS>L.-E.
Persson, UiT The Arctic University of Norway, P.O. Box 385, N-8505, Narvik,
Norway and Department of Mathematics and Computer Science, Karlstad
University, 65188 Karlstad, Sweden<EMAIL_ADDRESS>G. Tephnadze, The
University of Georgia, School of science and technology, 77a Merab Kostava St,
Tbilisi 0128, Georgia<EMAIL_ADDRESS>
###### Abstract.

In the first part we review some crucial results concerning
convergence/divergence of Fourier series and the Féjer means for various
systems. As one new result we prove by concrete construction that for any set
$E$ of measure zero there exists a function $f\in L_{1}(G_{m})$ such that
Féjer means with respect to Vilenkin systems diverge on this set. We use new
constructions of Vilenkin polynomials, which was introduced in [22].

The research was supported by Shota Rustaveli National Science Foundation
grant no. FR-21-2844.

2000 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Fourier analysis, Vilenkin system, Vilenkin group,
Vilenkin-Fourier series, almost everywhere convergence, Carleson-Hunt theorem,
Kolmogorov theorem.

##  1\. Introduction

The almost-everywhere convergence of Fourier series for $L_{2}$ functions was
postulated by Luzin [19] in 1915 and the problem was known as Luzin’s
conjecture. Carleson’s theorem is a fundamental result in mathematical
analysis establishing the pointwise (Lebesgue) almost everywhere convergence
of Fourier series of $L_{2}$ functions, proved by Carleson [5] in 1966. The
name is also often used to refer to the extension of the result by Hunt [12],
which was given in 1968 to $L_{p}$ functions for $p\in(1,\infty)$ (also known
as the Carleson-Hunt theorem). Carleson’s original proof is exceptionally hard
to read, and although several authors have simplified the arguments there are
still no easy proofs of his theorem. Fefferman [7] published a new proof of
Hunt’s extension, which was done by bounding a maximal operator of partial
sums $S^{\ast}f:=\sup_{n\in\mathbb{N}}\left|S_{n}f\right|.$ This, in its turn,
inspired to derive a much simplified proof of the $L_{2}$ result by Lacey and
Thiele [18], explained in more detail in Lacey [17]. In the books Fremlin [8]
and Grafakos [9] were also given proofs of Carleson’s theorem. Already in
1923, Kolmogorov [16] showed that the analogue of Carleson’s result for
$L_{1}$ is false by finding such a function whose Fourier series diverges
almost everywhere (improved slightly in 1926 to diverging everywhere).

The analogue of Carleson’s theorem for Walsh system was proved by Billard [2]
for $p=2$ and by Sjölin [29] for $1<p<\infty$, and for bounded Vilenkin
systems by Gosselin [10]. Schipp [25] investigated the so called tree
martingales generalized the results about the maximal function, quadratic
variation and martingale transforms to these martingales. Using these results,
he gave a proof of Carleson’s theorem for Walsh-Fourier series. A similar
proof for bounded Vilenkin systems can be found in Schipp and Weisz [28] (see
also [33]). A proof of almost everywhere convergence of Walsh-Fourier series
was also given by Demeter [6] in 2015. Moreover, the corresponding result for
Vilenkin-Fourier series was proved in [22] (see also [23]) in 2022. Stein [30]
constructed an integrable function whose Walsh-Fourier series diverges almost
everywhere. Later Schipp [24] (see also [27]) proved that there exists an
integrable function whose Walsh-Fourier series diverges everywhere. Kheladze
[13, 14] proved that for any set of measure zero there exists a function in
$f\in L_{p}(G_{m})$ $(1<p<\infty)$ whose Vilenkin-Fourier series diverges on
the set, while the corresponding result for continuous or bounded function was
proved by Harris [11] or Bitsadze [3]. Another resent proof was given in the
new book [22].

In the one-dimensional case the weak type inequality for the maximal operator
of maximal operator of Fejér means

$\mu\left(\sigma^{*}f>\lambda\right)\leq\frac{c}{\lambda}\left\|f\right\|_{1}\text{
\qquad}\left(\lambda>0\right)$

can be found in Zygmund [35] for trigonometric series, in Schipp [26] for
Walsh series and in Pál, Simon [20] for bounded Vilenkin series. It follows
(see [22]) that the Fejér means $\sigma_{n}f$ of the Vilenkin-Fourier series
of $f\in L_{1}$ converge to $f$ almost everywhere.

Bugadze [4] proved that for any set of measure zero there exists a function in
$f\in L_{1}(G_{m})$ whose Féjer means of the Walsh-Fourier series diverges on
the set, while the corresponding result for the Vilenkin system was proved by
Karagulyan [15]. In particular, Karagulyan considered sequences of linear
operators $\\{U_{n}f(x)\\}$ with some localization property, which is
fulfilled for all summability methods with respect to orthonormal systems and
proved that for any set $E$ of measure zero there exists a set $A$ for which
$\\{U_{n}I_{A}(x)\\}$ diverges at each point $x\in E,$ where $I_{A}$ is a
characteristic function of a subset $A.$ This result is a generalization of
analogous theorems known for the Fourier sum operators with respect to
different orthogonal systems.

In this paper we construct explicit function in $f\in L_{1}(G_{m})$ whose
Féjer means with respect to Vilenkin system diverges on any set of measure
zero.

##  2\. Preliminaries

Denote by $\mathbb{N}_{+}$ the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$ Denote by
$Z_{m_{k}}:=\\{0,1,...,m_{k}-1\\}$ the additive group of integers modulo
$m_{k}$.

Define the group $G_{m}$ as the complete direct product of the groups
$Z_{m_{i}}$ with the product of the discrete topologies of $Z_{m_{j}}`$s. The
elements of $G_{m}$ are represented by sequences

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

The direct product $\mu$ of the measures
$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \ }(j\in Z_{m_{k}})$ is the Haar
measure on $G_{m}$ with $\mu\left(G_{m}\right)=1.$

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

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

where $x\in G_{m},$ $n\in\mathbb{N}.$ Denote $I_{n}:=I_{n}\left(0\right)$ for
$n\in\mathbb{N}_{+},$ and $\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$.

If we define the so-called generalized number system based on $m$ by

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

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 numbers of $n_{j}`$s differ from
zero. For two natural numbers $n=\sum_{j=1}^{\infty}n_{j}M_{j}$ and
$k=\sum_{j=1}^{\infty}k_{j}M_{j}$, we define the operation $\oplus$ by
$n\oplus
k:=\sum_{i=0}^{\infty}\left(\left(n_{j}+k_{j}\right)\pmod{m_{i}}\right){M_{j}},\quad
n_{j},k_{j}\in Z_{m_{j}}.$ It is known that (see [22])

$\psi_{n\oplus
k}\left(x\right)=\psi_{k}\left(x\right)\psi_{n}\left(x\right),\text{\ \
}\left(k,n\in\mathbb{N},\text{ \ \ }x,y\in G_{m}\right).$

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

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

Now, define the Vilenkin system (see [31]) $\psi:=(\psi_{n}:n\in\mathbb{N})$
on $G_{m}$ as:

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

Specifically, we call this system the Walsh-Paley system when $m\equiv 2.$ The
Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)$ (see
e.g. the books [1] and [27]).

If $f\in L_{1}\left(G_{m}\right)$, we define the Fourier coefficients, partial
sums of the Fourier series, Dirichlet kernels with respect to the Vilenkin
system by

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

respectively. Recall that (see e.g. [1] and [22])

(1)
$D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{ll}M_{n},&\text{if\thinspace\thinspace\thinspace}x\in
I_{n},\\\ 0,&\text{if}\,\,x\notin I_{n}.\end{array}\right.$

The Féjer means and kernels with respect to Vilenkin and systems are
respectively defined by

$\sigma_{n}f:=\frac{1}{n}\sum_{k=0}^{n-1}S_{k}f,\ \ \
K_{n}:=\frac{1}{n}\sum_{k=0}^{n-1}D_{k}=\sum_{k=0}^{n-1}\left(1-\frac{k}{n}\right)\psi_{k},\text{
\ \ }\left(n\in\mathbb{N}_{+}\right).$

We also define the maximal operators $S^{\ast}$ and $\sigma^{\ast}$ of the
partial sums and Féjer means with respect to Vilenkin and trigonometric
systems, respectively by

$S^{\ast}f:=\sup_{n\in\mathbb{N}}\left|S_{n}f\right|\ \ \ \ \ \text{and}\ \ \
\ \ \sigma^{\ast}f:=\sup_{n\in\mathbb{N}}\left|\sigma_{n}f\right|.$

##  3\. Almost everywhere divergence of Vilenkin-Fourier series

Our main theorem reads:

######  Theorem 1.

If $E\subseteq G_{m}$ is a set of measure zero, then there exists a function
$f\in L_{1}(G_{m})$ such that Féjer means with respect to Vilenkin systems
diverge on this set.

###### Proof.

We begin with a general remark. If $A\subseteq G_{m}$ is a finite union of
intervals $I_{1},I_{2},\dots,I_{n}$ for some $n\in\mathbb{N}_{+}$ and if $N$
is any non-negative integer, then there exists a Vilenkin polynomial $P$ such
that, for some $i\geq N,$ $P=\sum_{k=M_{N}}^{M_{i}-1}c_{k}\psi_{k},$ which
satisfies

$\left|P(x)\right|=1,\ (x\in A)\ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \
\int_{G_{m}}\left|P\right|^{p}d\mu=\mu(A).$

Indeed, if $i:=\max\\{M_{N},1/\mu(I_{j}):1\leq j\leq n\\},$ then, in view of
(1), we find that $P:=\chi(A)\psi_{i}$ is such a polynomial.

To prove the theorem, suppose $E\subseteq G_{m}$ satisfies that $\mu(E)=0.$
Cover $E$ with intervals $(I_{k},k\in\mathbb{N})$ such that
$\sum_{k=0}^{\infty}\mu(I_{k})<1$ and each $x\in E$ belongs to infinitely many
of the sets $I_{k}.$ Set $n_{0}:=0$ and choose integers
$n_{0}<n_{1}<n_{2}\dots$ such that
$\sum_{k=n_{j}}^{\infty}\mu(I_{j})<M_{j}^{-1}\ \ (j\in\mathbb{N}).$ Apply the
general remark above successively to the sets

$A_{j}:=\bigcup_{k=n_{j}}^{n_{j+1}-1}I_{k}\ \ \ \ \ \ (j\in\mathbb{N})$

to generate integers $\alpha_{0}:=0<\alpha_{1}<\alpha_{2}<\dots$ and Vilenkin
polynomials $P_{0},P_{1},\dots$ such that
$sp(P_{j})\subset\left[M_{\alpha_{j}},M_{\alpha_{j+1}}\right):$

$P_{j}=\sum_{k=M_{\alpha_{j}}}^{M_{\alpha_{j+1}}-1}c_{k}^{j}\psi_{k},$ (2)
$\|P_{j}\|_{p}^{p}=\mu(A_{j})\leq M_{j}^{-1}$

and

(3) $\left|P_{j}(x)\right|=1\ \ \ x\in A_{j},\ \ \ \text{for}\ \ \
j\in\mathbb{N}.$

Let

$g:=\sum_{j=1}^{\infty}P_{j}$

and choose integers $\beta_{0}<\beta_{1}<\beta_{2}<\ldots,$ such that
$\alpha_{j}<\beta_{j},\ j\in\mathbb{N}$ and

(4)
$M^{3}_{\beta_{j}}<\left(M_{\alpha_{j}}+M_{\beta_{j}}\right)^{3}<M_{\alpha_{j}}+M_{\beta_{j+1}}<M_{\alpha_{j+1}}+M_{\beta_{j+1}}<2M_{\beta_{j+1}}.$

Setting

$f:=\sum_{j=1}^{\infty}\psi_{M_{\beta_{j+1}}}P_{j}$

we observe by (2) that this series converges in $L_{p}(G_{m})$ norm. Hence
$f\in L_{p}(G_{m})$ and this series is the Vilenkin-Fourier series of $f.$
Moreover, since the spectra of the polynomials $P_{j}$ are pairwise disjoint
and

$\widehat{f}\left(k\right)=\left\\{\begin{array}[]{l}\text{ }c_{k}^{l},\text{
\ if \
}k=M_{\beta_{l+1}}+M_{\alpha_{l}},\ldots,M_{\beta_{l+1}}+M_{\alpha_{l+1}}-1,\
l\in\mathbb{N}_{+},\\\ \text{ }0,\text{ otherwise.}\end{array}\right.$

for any $j\in\mathbb{N}_{+}$ we have that

$\displaystyle\ \
\sigma_{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}f-\sigma_{M_{\alpha_{j}}+M_{\beta_{j+1}}}f$
$\displaystyle=$
$\displaystyle\psi_{M_{\beta_{j+1}}}P_{j}-\sum_{k=M_{\alpha_{j}}+M_{\beta_{j+1}}}^{M_{\alpha_{j+1}}+M_{\beta_{j+1}}-1}\frac{k}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}c^{j}_{k}\psi_{k}$
$\displaystyle+$
$\displaystyle\sum_{l=0}^{j-1}\sum_{k=M_{\alpha_{l}}+M_{\beta_{l+1}}}^{M_{\alpha_{l+1}}+M_{\beta_{l+1}}}\left(\frac{k}{M_{\alpha_{j}}+M_{\beta_{j+1}}}-\frac{k}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}\right)c^{l}_{k}\psi_{k}$
$\displaystyle:=$ $\displaystyle I-II+III.$

Since $\left|c^{j}_{k}\right|<C,$ for any $j\in\mathbb{N},$ according to (4)
for $II$ we find that

$\displaystyle|II|$ $\displaystyle\leq$ $\displaystyle
C\sum_{k=0}^{M_{\alpha_{j+1}}+M_{\beta_{j+1}}-1}\frac{k}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}\leq\frac{C\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{2}}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}$
$\displaystyle\leq$
$\displaystyle\frac{C}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)}\to 0,\
\text{as}\ j\to\infty.$

Analogously, if we apply again (4) we get that

$\displaystyle|III|$ $\displaystyle\leq$
$\displaystyle\sum_{l=0}^{j-1}\sum_{k=M_{\alpha_{l}}+M_{\beta_{l+1}}}^{M_{\alpha_{l+1}}+M_{\beta_{l+1}}}\left(\frac{k}{M_{\alpha_{j}}+M_{\beta_{j+1}}}-\frac{k}{\left(M_{\alpha_{j+1}}+M_{\beta_{j+1}}\right)^{3}}\right)|c^{l}_{k}|$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=0}^{M_{\alpha_{j}}+M_{\beta_{j}}}\frac{k}{M_{\alpha_{j}}+M_{\beta_{j+1}}}\leq
C\sum_{k=0}^{2M_{\beta_{j}}}\frac{k}{M_{\beta_{j+1}}}$ $\displaystyle\leq$
$\displaystyle\frac{C\left(M_{\beta_{j}}\right)^{2}}{M_{\beta_{j+1}}}\to 0,\ \
\text{as}\ \ j\to\infty.$

Since every $x\in E$ belongs to infinitely many of the sets $A_{j}$ it follows
from (3) that

$|I|=1,\ \text{for }\ x\in I_{A_{j}},\ \text{for infinitely many }\ j.$

Hence, we obtain that the Vilenkin-Fourier series of $f$ diverges at every
point $x\in E.$ The proof is complete. ∎

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