G. Tephnadze
In this paper we investigate convergence and strong summability of the two-dimensional Vilenkin-Fourier series in the martingale Hardy spaces.
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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…
In this paper, we investigate very general approximation kernels with special properties, called an approximate identity, and prove almost everywhere and norm convergence of these general methods, which consists of a class of summability methods and provide norm and a.e. convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fejér means with respect to the Vilenkin system also, but these methods are not useful to study a.e. convergence in this case, because of some special properties of the…
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 $L^{p}$.
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.
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 $0<p<1.$
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.
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.
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.
In this paper we investigate strong summability of the two-dimensional Walsh-Fourier series obtained in Weisz \cite{We} (see Theorem W) and prove sharpness of this result.
We consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fej{é}r and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space $H_p$ to $L_p$, for all $p>p_0$, where $p_0$ depends on the summability method and the dimension. For $p=1$, we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means. Similar results are formulated for the more genera…
The main aim of this paper is to investigate Paley type and Hardy-Littlewood type inequalities and strong convergence theorem of partial sums of Vilenkin-Fourier series.
The main aim of this paper is to find the estimation for Vilenkin-Fourier coefficients.
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.
The main aim of this paper is to investigate (H_{p},L_{p})-type inequalities for maximal operators of logarithmic means of one-dimensional Vilenkin-Fourier series.
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.
TThe main aim of this paper is to prove that there exist a martingale $f\in H_{1/2}$ such that Fejér means of Vilenkin-Fourier series of the martingale $f$ is not uniformly bounded in the space $L_{1/2}.$
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…
In this paper we shall be concerned with $H_α$ summability, for $0 < α\leq 2$ of the Fourier series of arbitrary $L^1([-π,π]) $ functions. The methods to be employed here are a refinement of the real variable methods introduced by Marcinkiewicz in \cite{Marcin1}. In addition, we introduce maximal theorems with respect to the Lebesgue measure and $A_1$ weights.
An analogue of D. Waterman's result on the summability of the Fourier series for functions of bounded Λ-variation by the Cesáro methods of negative order is obtained in multidimensional case. It is proved that, unlike one-dimensional case, the continuity of function in the corresponding variation is essential for the convergence and even for the localization of the Cesáro means for certain orders of these means.
The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces $H_{p},$ when $0<p\leq 1.$
The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.
We study the $\ell^1$-summability of functions in the $d$-dimensional torus $\mathbb{T}^d$ and so-called $\ell^1$-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the $\ell^1$-norm of their indices. Such functions are characterized as divided differences that have $\cos θ_1,\ldots,\cosθ_d$ as knots for $(θ_1\,\ldots, θ_d) \in \mathbb{T}^d$. It leads us to consider the $d$-dimensional Fourier series of univariate B-splines with respect to its knots, which turns out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal series o…
We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $L^p(\mathbb T^d)$, $1<p<2$, we consider Riesz means at the critical index $d(1/p-1/2)-1/2$ and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal op…
Pointwise summability of Fourier-Laguerre series of integrable functions
本文给出了关于可积函数的Fourier-Laguerre级数点态可和性的逼近版本结果，并附有推论和例子。
We establish two general theorems on the local properties of the absolute summability of factored Fourier series by applying a recently defined absolute summability, $\left\vert A,α_{n}\right\vert _{k}$ summability, and the class $\mathcal{S}\left( α_{n},ϕ_{n}\right) $, which generalize some well known results and can be applied to improve many classical absolute summability methods.
The properties of the maximal operator of the $(C,α)$-means ($α=(α_1,\ldots,α_d)$) of the multi-dimensional Walsh-Kaczmarz-Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from $H_p^γ$ to $L_p$ for $p_0< p $ ($p_0=\max{1/(1+α_k): k=1,\ldots, d}$) and is of weak type $(1,1)$. As a corollary we get the theorem of Simon \cite{S1} on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. At the endpoint case $p=p_0$, we show that the maximal operator $σ^{κ,α,*}_L$ is not bounded fro…
The following problem is studied in this paper: Which multipliers $\{λ_{k, n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-π, π]$ $$ \sum_{k=-\infty}^\infty λ_{k, n}\hat{f}_k e^{ikx}, $$ where $\widehat{f}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function $\int_0^x f$ exists. A criterion for the convergence of the $(C, 1)$-means ($λ_{k, n}=(1-\frac {|k|}{n+1})_+$) is found, while in the general case $λ_{k, n}=ϕ(\frac {k}{n+1})$ a sufficient condition is derived for the convergence at all su…
Cesàro Summability of Fourier Orthogonal Expansions on the Cylinder
研究圆柱上基于正交多项式的傅里叶正交展开在L^p范数下的Cesàro可和性，得到了临界指标δ的上界。
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 theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.
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 $Φ(t):[0,\infty)\to[0,\infty)$ is bigger than the exponent, then the strong $Φ$-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.
We prove a Korovkin type approximation theorem via power series methods of summability for continuous $2π$-periodic functions of two variables and verify the convergence of approximating double sequences of positive linear operators by using modulus of continuity. An example concerning double Fourier series is also constructed to illustrate the obtained results.
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.
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。
Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality $\sum_{k}|f_k|^2θ^k<\infty$ to hold with $θ>1$. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of reproducing kernel Hilbert spaces.
Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that allow an extensive theory for Fourier expansions and approximation.
The space $\widehat{\ell }_k$ of absolutely almost convergent series was introduced and studied by Das et al [4]; which plays an important role in summability theory, approximation theory, Fourier analysis, etc. In the present paper we generalize the space making use of some factors and weighted mean transformations, investigate its topological structures and relations between classical sequence spaces. Also we characterize certain matrix operators on it.
The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. 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 p…
Let $f \in L^1(\mathbb{R}^2)$ and let $\widehat f$ be its Fourier integral. We study summability of the partial integral $S_{ρ,\mathsf{H}}(x)=\int_{\{\|y\|_\mathsf{H} \le ρ\}} e^{i x\cdot y}\widehat f(y) dy$, where $\|y\|_\mathsf{H}$ denotes the uniform norm taken over the regular hexagonal domain. We prove that the Riesz $(R,δ)$ means of the inverse Fourier integrals are nonnegative if and if $δ\ge 2$. Moreover, we describe a class of $\|\cdot\|_\mathsf{H}$-radial functions that are positive definite on $\mathbb{R}^2$.
We consider summability methods generated by the class GM(2b). We generalize some related results of P. Pych-Taberska [Studia Math. XCVI (1990), 91-103] on strong approximation of almost periodic functions by their Fourier series and S. M. Mazhar and V. Totik [J. Approx. Theory, 60(1990), 174-182] on approximation of periodic functions by matrix means of their Fourier series.
We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H; Phi) as an approximation version of the Totik type generalization of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from L^{Psi}. As a measure of such approximation we will use the function constructed by function Psi complementary to Phi on the base of defnition of the L^{Psi} points. Some corollary and remarks will also be given.
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。
A family of convolution operators, part two
本文研究一族 Fourier 乘子由光锥奇性分布定义的卷积算子，并据此提出 Bochner-Riesz 可和性的新方法。
Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $δ>0$, and we obtain estimates for other pairs of spaces for larger values of $δ$. Our study is broad enough to encompass general bilinear multipliers $m(ξ,η)$ radial in $ξ$ and $η$ with minimal smoothness, measured in Sobolev space norms. Our…
Given an irrational vector $α$ in $\mathbb{R}^{d}$, a continuous function $f(x)$ on the torus $\mathbb{T}^{d}$ and suitable weights $Φ(N,n)$ such that $\sum_{n=-\infty}^{+\infty}Φ(N,n)=1$, we estimate the speed of convergence to the integral $\int_{\mathbb{T}^{d}}f(y)dy$ of the weighted sum $\sum_{n=-\infty}^{+\infty}Φ(N,n) f(x+nα)$ as $N\rightarrow +\infty$. Whereas for the arithmetic means $N^{-1}\sum_{n=1}^{N}f(x+nα)$ the speed of convergence is never faster than $cN^{-1}$, for other means such speed can be accelerated. We estimate the speed of convergence in two theorems with different fla…
A new look at old theorems of Fejér and Hardy
本文通过抽象赋范空间的新 Tauberian 定理研究三角 Fourier 级数收敛性，给出弱于 Hardy 条件的充分条件，并推广了经典 Fejér 和 Hardy 定理。
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under the Alexiewicz norm, $\|f\|_\T =\sup_{|I|\leq 2π}|\int_I f|$, the supremum being taken over intervals of length not exceeding $2π$. It contains the periodic functions integrable in the sense of Lebesgue and Henstock-Kurzweil. Many of the properties of $L^1$ Fourier series continue to hold for this larger space, with the $L^1$ norm replaced by the Alexiewicz …
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 …
We develop a local theory of lacunary Dirichlet series of the form $\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the boundary $i\RR$, under the assumption $g'\to\infty$ and further assumptions on $c_k$. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with $c_k=1$, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at $z=0$. When sufficient analyticity information on $g…
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