On the partial sums of Vilenkin-Fourier series

On the strong convergence of partial sums with respect to bounded Vilenkin systems

In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.

On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

Convergence and Strong Summability of the Two-dimensional Vilenkin-Fourier Series

In this paper we investigate convergence and strong summability of the two-dimensional Vilenkin-Fourier series in the martingale Hardy spaces.

Some inequalities for Cesàro Means of double Vilenkin-Fourier Series

In this paper we state and prove some new inequalities related to the rate of $L^{p}$ approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from $L^{p}$.

Strong convergence of two--dimensional Vilenkin-Fourier series

We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p\leq 1.$ We also prove that the sequence in the denominator cannot be improved.

A note on The maximal operators of the Nörlund logaritmic means of Vilenkin-Fourier series

The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for the the maximal operators of Nörlund logaritmic means, for $0<p<1.$

On the Nörlund means of Vilenkin-Fourier series

In this paper we prove and discuss some new $\left( H_{p},L_{p}\right)$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients. These results are the best possible in a special sense. As applications, both some well-known and new results are pointed out in the theory of strong convergence of Vilenkin-Nörlund means with non-increasing coefficients.

On the convergence of Cesáro means of negative order of Vilenkin-Fourier series

In 1971 Onnewer and Waterman establish sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In the paper we consider different classes of functions of generalized bounded oscilation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesáro means of negative order.

A sharp boundedness result for restricted maximal operators of Vilenkin-Fourier series on martingale Hardy spaces

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.

Some new resuts on strong convergence of Fejér means with respect to Vilenkin systems

In this paper we discuss and prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system.

Ph.D. Thesis-Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

In this PhD thesis we discuss, develop and apply this fascinating theory connected to modern harmonic analysis. In particular we make new estimations of Vilenkin-Fourier coefficients and prove some new results concerning boundedness of maximal operators of partial sums. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of the partial sums is valid and develop new methods to prove Hardy type inequalities for the partial sums with respect to the Vilenkin systems. We also do the similar investigation for the Fejér means. Furthermore, we…

On The maximal operators of Vilenkin-Fejér means

The main aim of this paper is to prove that the maximal operator $\overset{% \sim }{σ}^{*}f:=\underset{n\in P}{\sup }\frac{\left| σ_{n}f\right| }{\log ^{2}\left( n+1\right) }$ is bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}$, where $σ_{n}f$ is Fejér means of bounded Vilenkin-Fourier series.

Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions

In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in $L_p$ norms of such Nörlund means.

Lebesgue and Vilenkin-Lebesgue points and a.e. Convergence of T means with respect to Vilenkin systems of integrable functions

In this paper we derive converge of $T$ means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in $L_p$ norms of such $T$ means.

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

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 p…

关于零测集上Vilenkin系统的Féjer平均的发散性

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

该研究针对零测集上Vilenkin系统Féjer平均的发散性问题，通过新的Vilenkin多项式构造，证明任意零测集上存在L_p函数使其Féjer平均发散，给出了显式构造的替代证明方法。

Some Inequalities Related to Strong Convergence of Riesz Logarithmic Means

In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series.

An Application of Fourier Analysis to Riemann Sums

We develop a method for calculating Riemann sums using Fourier analysis.

An exponential estimate for the square partial sums of multiple Fourier series

We prove an exponential integral estimate for the quadratic partial sums of multiple Fourier series on large sets that implies some new properties of Fourier series.

On the divergence of triangular and eccentrical spherical sums of double Fourier series

We construct a continuous function on the torus with almost everywhere divergence triangular sums of double Fourier series. An analogous theorem we also prove for eccentrical spherical sums.

Approximations of classes of generalized Poisson integrals by Fourier sums in metrics of spaces $L_{s}$

In metrics of spaces $L_{s}, \ 1\leq s\leq\infty$, we find asymptotic equalities for upper bounds of approximations by Fourier sums on classes of generalized Poisson integrals of periodic functions, which belong to unit ball of space $L_{1}$.

On the partial sums of Walsh-Fourier series

In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of convergence of the partial sums of the Walsh series with the modulus of continuity of martingale. These conditions are in a sense necessary and sufficient.

Estimates for approximations by Fourier sums, best approximations and best orthogonal trigonometric approximations of the classes of (ψ, β)-differentiable functions

We obtain the exact-order estimates for approximations by Fourier sums, best approximations and best orthogonal trigonometric approximations in metrics of spaces L_s, 1\leq s<\infty, of classes of 2π-periodic functions, whose (ψ,β)-derivatives belong to unit ball of the space L_\infty.

Indicator functions with uniformly bounded Fourier sums and large gaps in the spectrum

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term "Fourier sums" should be understood as "partial Fourier integrals". A certain weighted version of the result is also provided. This version leads to a new Men$'$shov-type correction theorem.

Lebesque-type inequalities for the Fourier sums on classes of generalized Poisson integrals

For functions from the set of generalized Poisson integrals $C^{α,r}_βL_{p}$, $1\leq p <\infty$, we obtain upper estimates for the deviations of Fourier sums in the uniform metric in terms of the best approximations of the generalized derivatives $f^{α,r}_β$ of functions of this kind by trigonometric polynomials in the metric of the spaces $L_{p}$. Obtained estimates are asymptotically best possible.

Uniform approximations by Fourier sums on classes of generalized Poisson integrals

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2π$-periodic functions, representable in the form of convolutions of functions $φ$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.

Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness

We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{β,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{β,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.

Uniform approximations by Fourier sums on classes of convolutions of periodic functions

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2π$-periodic functions, which are represented by convolutions of functions $φ(φ\bot 1)$ from unit ball of the space $L_{1}$ with fixed kernels $Ψ_β$ of the form $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k) \cos\left(kt-\frac{βπ}{2}\right)$, $\sum\limits_{k=1}^{\infty}kψ(k)<\infty$, $ψ(k)\geq 0$, $β\in\mathbb{R}$.

Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals

In this paper we establish Lebesgue-type inequalities for $2π$-periodic functions $f$, which are defined by generalized Poisson integrals of the functions $φ$ from $L_{p}$, $1\leq p< \infty$. In these inequalities uniform norms of deviations of Fourier sums $\| f-S_{n-1} \|_{C}$ are expressed via best approximations $E_{n}(φ)_{L_{p}}$ of functions $φ$ by trigonometric polynomials in the metric of space $L_{p}$. We show that obtained estimates are asymptotically best possible.

Lebesgue类函数在满测集上傅里叶和收敛速度的估计

Estimate of the Rate of Convergence of Fourier Sums for Functions from Lebesgue Classes on a Set of Full Measure

本文估计了Lebesgue类函数三角傅里叶级数部分和在满测集上的收敛速度，并通过反例证明该估计阶的精确性，进而导出Lebesgue类函数二维三角傅里叶级数几乎处处Prinsheim收敛的连续性模条件。

Approximation by Fourier sums in classes of Weyl--Nagy differentiable functions with high exponent of smoothness

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2π$-periodic Weyl--Nagy differentiable functions, $W^r_{β,p}, 1\le p\le \infty, β\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge \sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le p\le\infty,$ for functional classes $W^r_{β,1}$.

Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness

On the sets of $2π$-periodic functions $f$, which are defined with a help of $(ψ, β)$-integrals of the functions $φ$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $φ$. We prove that obtained estimates are best possible, in the case when the sequences $ψ(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on …

Cosine products, Fourier transforms, and random sums

We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments.

Erdos-Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields

We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.

Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric

We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating kernel $Ψ_β$, whose absolute values of Fourier coefficients $ψ(k)$ are such that $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$, and product $ψ(n)n^{\frac{1}{p}}$ can't tend to nought faster than power functions.

Lebesgue点处Fourier部分和多项式子序列的Cesàro平均

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

该研究对Zalcwasser1936年提出的两个问题给出肯定答案，证明了Fourier部分和沿特定多项式子序列的(C,α)平均在L¹(𝕋)函数的所有Lebesgue点处逐点收敛到原函数。

A note on the strong convergence of two-dimensional Walsh-Fourier series

The main aim of this paper is to investigate the quadratical partial sums of the two-dimensional Walsh-Fourier series.

Order estimations of the best approximations and approximations of the Fourier sums on the classes of infinitely differentiable functions

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2π$-periodic continuous functions, which $(ψ,β)$-derivatives $f_β^ψ$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $ψ(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(ψ,β)$-differentiable functions, such that $\parallel f_β^ψ\parallel_{1}\leq1$.

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 $φ(L)[0,1)$.

Cesàro means of subsequences of partial sums of trigonometric Fourier series

In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system that how "rare" can a sequence of strictly monotone increasing integers $(n_j)$ be such that the almost everywhere relation $\frac{1}{N}\sum_{j=1}^N S_{n_j}f \to f$ is fulfilled for each integrable function $f$. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function $f$ and lacunary sequence $(n_j)$ of natural numbers.

广义泊松积分类上傅里叶和的逼近

Approximation by Fourier sums on the classes of generalized Poisson integrals

本文综述了广义泊松积分类$C^{α,r}_{β,p}$上傅里叶和的Kolmogorov–Nikolsky问题的研究结果，明确了该问题的目标及对应函数类的定义。

Strong convergence of two-dimensional Walsh-Fourier series

We prove that certain mean of the quadratical partial sums of the two-dimensional Walsh-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p<1.$

Pointwise Convergence of Dyadic Partial Sums of Almost Periodic Fourier Series

It is a classical result that dyadic partial sums of the Fourier series of functions $f \in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1, \infty)$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operator corresponding to dyadic partial summation of almost periodic Fourier series is bounded on the Stepanov spaces $S^{2^k}$, $k \in \mathbb{N}$ for functions satisfying the same condition; Bredihina's result follow…

Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics

In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in\mathbb{R}$, whose coefficients $ψ(k)$ are such that product $ψ(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}ψ^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_…

Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(ψ,β)$--diferentiable functions

There were established the exact-order estimations of the best uniform approximations byψ the trigonometrical polynoms on the $C^ψ_{β,p}$ classes of $2π$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $Ψ_β\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^ψ_{β,1}$ classes of $2π$-periodic fu…

On the Partial Sums and Marcinkiewicz and Fejér Means on the One- and Two-dimensional One-parameter Martingale Hardy Spaces

In this PhD thesis we are dealing with convergence and summability of partial sums, Fejér and Marcinkiewicz means with respect to one- and two-dimensional Walsh-Fourier series on the martingale Hardy spaces. This thesis is focus to achieve the following main results: To find estimation of convergence and divergence of the subsequences of partial sums of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_p(G)$, when $0<p\leq1$. To find necessary and sufficient conditions in terms of modulus of continuity of martingale Hardy spaces, for which subsequences of partial sums …

Walsh-Paley傅里叶部分和的子序列的Cesàro均值几乎处处发散

Almost everywhere divergence of Cesaro means of subsequences of Walsh--Paley Fourier partial sums

证明存在严格递增正整数序列和L^1函数，使得Walsh-Fourier部分和子序列的Cesàro均值几乎处处发散，揭示了二进结构下仍存在发散现象。

Weyl-Nagy类 $W^r_{β,1}$ 上 Fourier 和偏差的估计

Estimates of deviations of Fourier sums on Weyl-Nagy classes $W^r_{β,1}$

本文在Weyl-Nagy类上建立了部分Fourier和偏差的精确上界估计，并由此得到参数n与r以任意关系趋于无穷时的渐近等式。

具有Lipschitz类导数函数的Cesáro广义Fourier和的性质

The Properties of Cesáro General Fourier Sums of Functions with Derivatives of Lipschitz Class Functions

本文研究导数属于Lipschitz类的函数关于一般正交函数系的Fourier级数的Cesáro平均，确定了保证其一致有界的系统函数条件，且结果最优。

Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\su…

Sparse Averages of Partial Sums of Fourier Series

We study convergence properties of sparse averages of partial sums of Fourier series of continuous functions. By sparse averages, we are considering an increasing sequences of integers $n_0 < n_1 < n_2 < ...$ and looking at \begin{equation*} \tildeσ_N(f)(t) = \frac{1}{N+1}\sum_{k=0}^{N}s_{n_k}(f)(t) \end{equation*} to determine the necessary conditions on the sequence $\{n_k\}$ for uniform convergence. Among our results, we find that convergence is dependent on the sequence: we give a proof of convergence for the linear case, $n_k = pk$, for $p$ a positive integer, and present strong experimen…

矩形傅里叶和的一个Oskolkov--Zhizhiashvili准则

An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

针对d维环面上三角傅里叶级数的对称矩形部分和，在Zhizhiashvili临界尺度下证明了可和端点准则，改进了经典充分条件。

三维和四维中的傅里叶插值及实变量 Kloosterman 和

Fourier interpolation in dimensions 3 and 4 and real-variable Kloosterman sums

本文利用 Maass--Poincaré 型级数构造三维和四维的径向傅里叶插值公式，给出基函数用实变量 Kloosterman 和的显式表达，并改进基函数关于指标的界。

Speed of convergence of two-dimensional Fourier integrals

Recently we found necessary and sufficient conditions for the convergence at a preassigned point of the spherical partial sums of the Fourier integral in a class of piecewise smooth functions in Euclidean space. These yield elementary examples of divergent Fourier integrals in three dimensions and higher. Meanwhile, several years ago Gottlieb and Orsag observed that in two dimensions we may expect slower convergence at certain points, specifically for Fourier-Bessel series of radial functions. In this paper we investigate the rate of convergence of the spherical partial sums of the Fourier int…