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Ushangi Goginava

Publications and source records attributed to Ushangi Goginava.

At least 19 recordsLinked to original sources
arXiv2026

A Counterexample to Belinsky's Conjecture on Cesàro Means at Lebesgue Points

In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\uı is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.

math.CA↗
arXiv2026

Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points

In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Cesàro means of fractional order. We give affirmative answers to both questions in a stronger pointwise form. Let $0<α\leq 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,α)$ means of the Fourier partial sums along any sequence whose $k$-th term equals $P(k)$ for all sufficiently large $k$ converges to $f(x)$ at every Lebesgue point $x$ of every $f\in L^{1}(\T)$.

math.CA↗
arXiv2026

A Critical-Scale Extension of Zhizhiashvili's Theorem for Rectangular Fourier Series

We address a long-standing endpoint problem arising from Zhizhiashvili's logarithmic modulus theorem for multiple Fourier series. We prove an endpoint Dini criterion for almost-everywhere Pringsheim convergence of ordinary rectangular partial sums. In Zhizhiashvili's theorem the logarithmic modulus is assumed with an exponent strictly above the critical value; here this strict power margin is replaced by a summable endpoint Dini condition. As a consequence, one obtains double-logarithmic endpoint classes lying outside the range of the classical theorem. The proof reduces the endpoint smoothness assumption to the Kaczmarz--Kojima product-logarithmic coefficient criterion by weighted translation-difference estimates.

math.CA↗
arXiv2026

An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

Let $S_{\mathbf n}f$ denote the symmetric rectangular partial sums of the trigonometric Fourier series of a function on the $d$-dimensional torus. We prove a summable endpoint criterion at the Zhizhiashvili critical scale for all $d\ge2$ and $1\le p\le2$. The criterion allows a general summable secondary weight at the iterated-logarithmic level and contains, as special cases, a double-logarithmic endpoint criterion and an $L^p$ Oskolkov-type corollary. In particular, it answers the Zhizhiashvili--Marcinkiewicz problem for $1<p<2$ and sharpens Zhizhiashvili's classical sufficient conditions in the endpoint cases $p=1$ and $p=2$.

math.CA↗
arXiv2026

Almost everywhere divergence of double Fourier series along shrinking conical regions

We study almost everywhere divergence of rectangular partial sums of double trigonometric Fourier series along variable regions concentrated near the diagonal. Fefferman's theorem shows that, in two variables, unrestricted rectangular summation is radically different from the one-dimensional Carleson-Hunt theory: there are continuous functions whose rectangular Fourier sums diverge everywhere. Bakhvalov proved that this phenomenon persists even when the indices are restricted to a fixed cone-shaped neighbourhood of the diagonal. On the other hand, the diagonal summation results of Tevzadze and Fefferman, and the later theorem of Antonov for shrinking cones, show that convergence is restored when the aperture is of order $1/n$. We prove that Antonov's aperture condition is sharp in the $L^{2}$-scale. For every positive nonincreasing sequence $\{λ_{n}\}$ with $% \sup_{n}nλ_{n}=\infty $, we construct a function $f\in L^{2}(\mathbb{T% }^{2})$ whose symmetric rectangular Fourier partial sums fail to converge almost everywhere along the corresponding variable cone. The construction combines a Fefferman-type analytic block with frequency separation and independent random translations. We also observe that the sufficiency part of Antonov's theorem does not require monotonicity of $\{λ_{n}\}$.

math.CA↗
arXiv2026

Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators

We study weighted Walsh--Carleson maximal operators arising from dyadic martingale transforms associated with Walsh--Fourier partial sums. For weights satisfying a uniform dyadic variation condition and a uniform bound at the top dyadic scale, we prove weak type~$(1,1)$ estimates for the corresponding maximal operators along subsequences. We also give divergence criteria in terms of the behavior of the weights near the top dyadic scale and, under suitable admissibility assumptions, relate these criteria to explicit ratio conditions. As applications, we obtain results on matrix transforms of Walsh--Fourier partial sums, including de la Vallée Poussin means, Cesàro means with varying parameters, Nörlund logarithmic means, and general Nörlund means. In particular, we prove a Walsh--Paley analogue of the Leindler--Tandori theorem and establish everywhere divergence results for several summability methods.

math.CA↗
arXiv2026

de la Vallée Poussin Means of Walsh-Fourier Expansions

We study de la Vallée Poussin means of Walsh--Fourier series associated with a nondecreasing window sequence. We establish a sharp criterion for almost everywhere convergence for integrable functions. We further show that, when this criterion fails, every Orlicz class below the logarithmic square-root scale contains a function whose de la Vallée Poussin means diverge everywhere.

math.CA↗
arXiv2026

Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along Subsequences

Let $S_m f$ denote the $m$-th partial sum of the Walsh-Fourier series of $f \in L^1$. For an increasing sequence $a=(a(n))_{n \geq 1}$ of positive integers, consider the arithmetic means $$ σ_N f:=\frac{1}{N} \sum_{n=1}^N S_{a(n)} f . $$ Gát proved in 2019 that $σ_N f \rightarrow f$ almost everywhere for every $f \in L^1$ under the growth condition $$ a(n+1) \geq\left(1+\frac{1}{n^δ}\right) a(n), \quad 0<δ<\frac{1}{2} . $$ We show that the same conclusion remains valid throughout the full range $0<δ<1$.

math.CA↗
arXiv2024

Limits of sequences of operators associated with Walsh System

The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by Gát and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.

math.AP↗
arXiv2024

Limits of sequence of Tensor Product Operators associated with the Walsh-Paley system

It is well-known that to establish the almost everywhere convergence of a sequence of operators on $L_1$-space, it is sufficient to obtain a weak $(1,1)$-type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.

math.AP↗
arXiv2019

On the divergence of subsequences of partial Walsh-Fourier sums

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 $φ:[0,\infty)\rightarrow[0,\infty)$ is given fulfilment of which implies an existence of above type function $f$ in the class $φ(L)[0,1)$.

math.AP↗
arXiv2019

Conjugate Transforms on Dyadic Group

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejér means we will find necessary and sufficient condition on $t$ for which the estimation $E\left\vert \widetilde{% σ}_{n}^{\left( t\right) }f\right\vert \lesssim E\left\vert f\right\vert $ holds . We also prove that for dyadic irrational $t$, $L\log L $ is maximal Orlicz space for which the estimation $E\left\vert \widetilde{% σ}_{n}^{\left( t\right) }f\right\vert \lesssim 1+E\left( \left\vert f\right\vert \log ^{+}\left\vert f\right\vert \right) $ is valid.

math.AP↗
arXiv2018

Logarithmic means of Walsh-Fourier Series

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.

math.AP↗