# Ushangi GoginavaIvane Javakhishvili Tbilisi State University | TSU · Department of Mathematics

Ushangi Goginava

Doctor of Sci

## About

203

Publications

5,984

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,483

Citations

Citations since 2016

Introduction

Ushangi Goginava currently works at the Department of Mathematics, Ivane Javakhishvili Tbilisi State University. Ushangi does research in Analysis. Their current project is 'strong summability of Fourier series'.

Additional affiliations

January 2006 - present

January 2003 - December 2012

**Tbilisi State University**

January 2001 - present

## Publications

Publications (203)

In this paper we discuss some convergence and divergence properties of subsequences of Cesàro means with varying parameters of Walsh–Fourier series. We give necessary and sufficient conditions for the convergence regarding the weighted variation of numbers.

In this paper we study the a.e. strong summability of the
quadratical and rectangular partial sums of the two-dimensional Vilenkin–Fourier series for every two-dimensional functions belonging to L log L. We
also study the exponential uniform strong approximation of Marcinkiewicz
type of two-dimensional Vilenkin–Fourier series.

The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L1 space and in CW space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t*(f) of...

It is proved that the maximal operators of subsequences of Cesàro means with varying parameters of two-dimensional Walsh-Fourier series is bounded from the dyadic Hardy spaces HpI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{up...

The presented paper will be proved the necessary and sufficient conditions in order maximal operator of Walsh-N\"orlund means with non-increasing weights to be bounded from the dyadic Hardy space $H_{p}(\mathbb{I})$\ to the space $% L_{p}(\mathbb{I})$.

In the present paper, we prove the almost everywhere convergence and divergence of subsequences of Cesàro means with zero tending parameters of Walsh–Fourier series.

It is proved that the maximal operators of subsequences of N?rlund logarithmic means and Ces?ro means with varying parameters of Walsh-Fourier series is bounded from the dyadic Hardy spaces Hp to Lp. This implies an almost everywhere convergence for the subsequences of the summability means.

A class of increasing sequences of natural numbers (nk) is found for which there exists a function f∈L[0,1) such that the subsequence of partial Walsh-Fourier sums (Snk(f)) diverges everywhere. A condition for the growth order of a function φ:[0,∞)→[0,∞) is given fulfilment of which implies an existence of above type function f in the class φ(L)[0,...

In this paper we study the a. e. exponential strong (C, 1, 0) summability of of the 2-dimensional trigonometric Fourier series of the functions belonging to L (log+L)2.

A class of increasing sequences of natural numbers $(n_k)$ is found for which there exists a function $f\in L[0,1)$ such that the subsequence of partial Walsh-Fourier sums $(S_{n_k}(f))$ diverge everywhere. A condition for the growth order of a function $\varphi:[0,\infty)\rightarrow[0,\infty)$ is given fulfilment of which implies an existence of a...

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fej\'er means we will find necessary and sufficient condition on $t$ for which the estimation $E\left\vert \widetilde{% \sigma }_{n}^{\left( t\right) }f\right\vert \lesssim E\left\vert f\right\vert $ holds . We also prove that for dyadic irrati...

In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means \(S_{2^A}^\Delta(f)\) of the...

The boundedness of maximal operators of subsequences of \((C,\alpha _{n})\)-means of partial sums of Walsh–Fourier series from the Hardy space Hp into the space Lp is studied.

In this paper we study the maximal operator for a class of subsequences of strong Nörlund logarithmic means of Walsh-Fourier series. For such a class we prove the almost everywhere strong summability for every integrable function f.

In this paper we study the criterions of the uniform convergence and L-convergence of double Vilenkin–Fourier series. We also prove that these results are sharp.

In this paper we discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series . We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers.

For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$ where n := ∑
i=0∞ni
2i, ni
∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supA
V (nA) < ∞, the subsequence of quadratic partial sums...

In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L p such that its triangular partial sums S △2A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S △n M f, n A ∈ {1, 2, …, m A − 1} on unbounded Vilenkin groups conv...

In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces X(G) for every f ∈ X(G), where by X(G) we denote either the class of continuous functions with supremum norm or the class of integrable functions.

In this paper we study the a. e. strong summability of the cubic partial sums of the d-dimensional Walsh-Fourier series of the functions belonging to L( log⁺ L)d-1.

In this paper we give a characterization of points at which the Marcinkiewicz-Fejér means of double Vilenkin-Fourier series converge.

In this paper we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of the functions from $L\log L$ .

It is proved a BMO-estimation for rectangular partial sums of two-dimensional Walsh-Fourier series from which it is derived an almost everywhere exponential summability of rectangular partial sums of double Walsh-Fourier series.

In this paper we study the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series. In particular, it is proved that the two-dimensional Vilenkin-Fourier series of the continuous function $f$ is uniformly strong summable to the function $f$ exponentially in the power $% 1/2$. Moreover, it is proved that this result is best...

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series of every continuous function f is uniformly strong summable to the function f exponentially in the power...

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fej\'er (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Na...

The main aim of this article is to prove that the maximal operator
of the Marcinkiewicz-Fejér means of the two-dimensional Fourier series with respect to Walsh-Kaczmarz system is bounded from the Hardy space H2/3 to the space weak-L2/3.

The sufficient and necessary conditions on the sequence Λ = {λn} are found for the uniformly convergence of Cesàro means of negative order of cubic partial sums of double Walsh-Fourier series of functions of bounded partial Λ-variation.

In this paper we prove a weak type \((1,1)\) inequality for the maximal operator \(\sup _{n,m}\frac{|\sigma _{n,m}|}{(n+m)^\varepsilon }\) for any \(\varepsilon >0\) . This allows us to reach some inequalities which hold almost everywhere. Moreover, we improve the result of Wade presented in (Proceedings of A. Haar memorial conference, 1987).

In this chapter, we present several open problems in theory and applications of dyadic derivatives and their generalizations. The problems are suggested by the contributors of this book.

F. Schipp in 1969 proved the almost everywhere p-strong summability of Walsh-Fourier series and showed that if λ(n)→∞, then there exists a function f ∈ L 1[0, 1) for which the Walsh partial sums S k (x, f) satisfy the divergence condition $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{1} {n}\sum\limits_{k = 1}^n {\left| {S_k (x,f)} \right|^{\l...

We prove inclusion relations between generalized Waterman's and generalized
Wiener's classes for functions of two variable.

In this paper we present results on convergence and Ces\`{a}ro summability of
Multiple Fourier series of functions of bounded generalized variation.

The convergence of partial sums and Ces\'aro means of negative order of
double Walsh-Fourier series of functions of bounded \ generalized variation is
investigated.

The maximal Orlicz spaces such that the mixed logarithmic means of multiple
Walsh-Fourier series for the functions from these spaces converge in measure
and in norm are found.

Almost everywhere strong exponential summability of Fourier series in Walsh
and trigonometric systems established by Rodin in 1990. We prove, that if the
growth of a function $\Phi(t):[0,\infty)\to[0,\infty)$ is bigger than the
exponent, then the strong $\Phi$-summability of a Walsh-Fourier series can fail
everywhere. The analogous theorem for trig...

Under study is the convergence of the negative order Cesàro means of the double trigonometric Fourier series of functions of bounded generalized Λ-variation.

The maximal Orlicz space such that the mixed logarithmic means of multiple
Fourier series for the functions from this space converge in $L_{1}$-norm is
found.

In this paper we study the a. e. strong convergence of the quadratical
partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the
a.e. relation $(\frac{1}{n}\sum\limits_{m=0}^{n-1}\left\vert S_{mm}f - f
\right\vert^{p})^{1/p}\rightarrow 0$ for every two-dimensional functions
belonging to $L\log L$ and $0<p\le 2$. From the theore...

It is proved that the operators $\sigma_{n}^{\bigtriangleup}$ of the
triangular-Fej{\'e}r-means of a two-dimensional Walsh--Fourier series are
uniformly bounded from the dyadic Hardy space $H_{p}$ to $L_{p}$ for all $%
4/5<p\leq \infty $.

The main aim of this article is to investigate the (H-p,L-p)-type inequality for the maximal operators of Riesz and Norlund logarithmic means of Walsh-Kaczmarz-Fourier series. Moreover, we show that the behaviour of Norlund logarithmic means is worse than the behaviour of Riesz logarithmic means in our special sense.

The main aim of this paper is to investigate the (H
p, L
p)-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.

It is proved a $BMO$-estimation for quadratic partial sums of two-dimensional
Fourier series from which it is derived an almost everywhere exponential
summability of quadratic partial sums of double Fourier series.

N\"orlund strong logarithmic means of double Fourier series acting from space
$% L\log L(\mathbb{T}^{2}) $ into space $L_{p}(\mathbb{T}% ^{2}), 0<p<1$ are
studied. The maximal Orlicz space such that the N\"o% rlund strong logarithmic
means of double Fourier series for the functions from this space converge in
two-dimensional measure is found.

The convergence of multiple Fourier series of functions of bounded partial $%
\Lambda$-variation is investigated. The sufficient and necessary conditions on
the sequence $\Lambda=\{\lambda_n\}$ are found for the convergence of multiple
Fourier series of functions of bounded partial $\Lambda$-variation.

We prove that certain means of the (C,α,…,α)-means (α=1/p−1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space H
p
to H
p
(1≦p≦2). As a consequence we obtain strong summability theorems concerning (C,α,…,α)-means.

In this paper we investigate some convergence and divergence properties of
the logarithmic means of quadratical partial sums of double Fourier series of
functions in the measure and in the $L$ Lebesgue norm.

The Uniform convergence of double Fourier-Legendre series of function of
bounded Harmonic variation and bounded partial $\Lambda $-variation are
investigated.

The paper introduces a new concept of $\Lambda $-variation of multivariable
functions and investigates its connection with the convergence of
multidimensional Fourier series

The problem of convergence of the Cesàro means of negative order for double Walsh–Fourier series of functions of bounded generalized variation is investigated.

The convergence of multiple Walsh-Fourier series of functions of bounded generalized variation is investigated. The sufficient and necessary conditions on the sequence λ = {λ
n
} are found for the convergence of multiple Walsh-Fourier series of functions of bounded partial λ-variation.

The paper introduces a new concept of Λ-variation of bivariate functions and investigates its connection with the convergence of double Fourier series.

It is proved that the maximal operator sigma(Delta)(#) of the triangular-Fejer-means of a two-dimensional Walsh-Fourier series is bounded from the dyadic Hardy space H-p to L-p for all 1/2 < p <= infinity and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejer-means sigma(Delta)(2n) of a function f is an eleme...

In the present paper we prove that for any 0 < p <= 2/3 there exists a martingale f in H-p such that the Marcinkiewicz-Fejer means of double conjugate Walsh-Kaczmarz-Fourier series of the martingale f is not uniformly bounded in the space L-p.

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of Llog + L(I 2 ), the set of the functions the logarithmic means of quadratical partial sums of the...

The convergence of Ces\`{a}ro means of negative order of double trigonometric
Fourier series of functions of bounded partial $\Lambda$-variation is
investigated. The sufficient and neccessary conditions on the sequence $\Lambda
=\{\lambda_{n}\}$ are found for the convergence of Ces\`{a}ro means of Fourier
series of functions of bounded partial $\La...

In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditionally in the dyadic Hardy space HM1 (G). We will do it for wavelets satisfying the regularity condition of Hölder-Lipshitz type.

It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the dyadic Hardy space H 1/2 to the space weak-L1/2/.

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H
p
such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L
p
.

In this paper we prove that the maximal operator
where σ
n
k
is the n-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space H
1/2(G) to the space L1/2(G).

In this paper we give a characterization of points in which Fejer means of Vilenkin-Fourier series converge.

In this paper we prove that the maximal operator I* of dyadic derivative is not bounded from the Hardy space Hp[0,1] to the Hardy space Hp[0,1], when 0 < p ≤ 1.

Define the two dimensional diagonal Sunouchi operator
where S 2 n , 2 n ƒ and σ 2 n ƒ are the (2 n , 2 n )th cubic-partial sums and 2 n th Marcinkiewicz–Fejér means of a two-dimensional Walsh–Fourier series. The main aim of this paper is to prove that the operator is bounded from the Hardy space H 1/2 to the weak L 1/2 space and is not bounded fro...

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh-Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh-Fourier series of the continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is...

In this paper we characterize the set of convergence of the Marcinkiewicz-Fejer means of two-dimensional Walsh-Fourier series.

In this paper we study the exponential uniform strong approximation of two-dimensional Walsh-Fourier series. In particular, it is proved that the two-dimensional Walsh-Fourier series of the continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let H-n(f, x, y, A(n)) := 1/n Sigma(n)(l=1) (e(An vertical bar Sll(f,x,y) - f(x,y)vertical bar 1/2) - 1). It is shown that if A(n) up arrow infinity arbitrary slowly, there exists f is an element of C(I-2) such that lim(n ->infinity) H-n (f, 0, 0, A(n)) = +inf...

We prove that certain means of the cubic partial sums of the two-dimensional
Walsh-Fourier series are uniformly bounded operator from dyadic Hardy space $%
H_{1}$ to the space $L_{1}$. As a consequence we obtain strong convergence
theorems concerning cubic partial sums.

The main aim of this paper is to prove that the maximal operator σ# is not bounded from the martingale Hardy space H
p
(G) to the martingale Hardy space H
p
(G) for 0 p ≤ 1.

Simon [J. Approxim. Theory,
127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H
p
to the space L
p
for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However,...

The main aim of this paper is to prove that there exists a martingale f is an element of H(1/2) such that the maximal Fejer operator with respect to Walsh-Kaczmarz system does not belong to the space L(1/2). For the two-dimensional case, we prove that there exists a martingale f is an element of H(1/2)(square)(f is an element of H(1/2)) such that t...

The main aim of this paper is to prove that the maximal operator σ
*α of the (C, α) means of the cubical partial sums of the two-dimensional Walsh-Fourier series is bounded from the Hardy space H
2/(2+α) to the space weak-L
2/(2+α).

Abstract: The main aim of this paper is to prove that for any 0 < p ≤ d/
/ (d + alpha) there exists a martingale f ∈ Hp such that the maximal operators of
(C, alpha) means of cubic partial sums of d-dimensional conjugate Walsh–Fourier
series do not belong to the space Lp.

The main aim of this paper is to prove that there exists a martingale f∈H 1/2 such that the maximal Fejér operator and the conjugate Fejér operator does not belong to the space L 1/2 .

The main aim of this paper is to prove that there exists a martingale f ∈ H
12/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier series does not belong to the space weak-L
1/2.

Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H
p
to the space L
p
for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operato...

The main aim of this paper is to prove that the maximal operator of Marcinkiewicz-Fejer means of double Fourier series with respect to the Walsh-Kaczmarz system is bounded from the dyadic Hardy-Lorentz space H(pq) into Lorentz space L(pq) for every p > 2/3 and 0 < q <= infinity. As a consequence, we obtain the a.e. convergence of Marcinkiewicz-Feje...

It is well known in the literature that the logarithmic means $$
\frac{1}
{{logn}}\sum\limits_{k = 1}^{n - 1} {\frac{{S_k (f)}}
{k}}
$$ of Walsh or trigonometric Fourier series converge a.e. to the function for each integrable function on the unit interval. This is not the case if we take the partial sums. In this paper we prove that the behavior o...

The main aim of this paper is to prove that the maximal operator σ# of the Marcin-kiewicz-Fejer means of the two-dimensional Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H1/2 to the space weak-L1/2 and is not bounded from the martingale Hardy space H1/2 to the space L 1/2 provided that the supr...