George Tephnadze
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.
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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…
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.
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.
In this paper we investigate convergence and strong summability of the two-dimensional Vilenkin-Fourier series in the martingale Hardy spaces.
We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for…
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…
We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.
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.
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.
We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_θ$ between another UMD space $X$ and a Hilbert space $H$, for some $θ\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.
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 Fejér (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence $σ_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$ ($n\to\infty$) for each integrable two-variable function $f$.
Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators
本文研究了由双侧马丁格尔变换产生的加权瓦尔什-卡拉森极大算子，证明了在满足均匀二进制变化条件和顶部二进制尺度上均匀界的情况下，这些算子在子序列上的弱型(1,1)估计，并给出了与权重在顶部二进制尺度附近行为相关的发散准则。
Approximation by a special de la Vallée Poussin type matrix transform mean of Walsh-Fourier series
该文研究Walsh系统下Fourier级数的特殊类de la Vallée Poussin矩阵平均的范数收敛，给出了用函数连续模表示的逼近误差上界。
We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{\frac12}}$ inequality is the core of…
In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left\vert σ_{k}F\right\vert/A^2_k\right)$ of Fejér means of Walsh-Fourier series and prove that the it is bounded from the martingale Hardy space $H_{1/2}(G)$ to the Lebesgue space $L_{1/2}(G).$ The sharpness of this result is also proved. As a consequence we obtain some new and and well-know results.
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…
In this paper we derive the maximal subspace of natural numbers $\left\{n_{k}:k\geq 0\right\}$, such that the restricted maximal operator, defined by $\sup_{k\in \mathbb{N}}\left\vert σ_{n_{k}}F \right\vert$ on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space $H_{1/2}$ to the Lebesgue space $L_{1/2}$. The sharpness of this result is also proved.
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…
For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result of Hunt-Young and is a weighted extension of a variation norm Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lepingle.
We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdor…
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) $, which generalize some well known results and can be applied to improve many classical absolute summability methods.
We prove a weak-$L^p$ bound for the Walsh-Carleson operator for $p $ near 1, improving on a theorem of Sjolin. We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function $f\in L\log L(\mathbb T)$ converge for almost every $x \in \mathbb T$.
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 …
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…
The main aim of this paper is to investigate the quadratical partial sums of the two-dimensional Walsh-Fourier series.
We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q<r \leq \infty and (r/(q-1))' < p < \infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t …
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。
We prove that Cesàro means of one-dimensional Walsh-Fourier series are uniformly bounded operators in the martingale Hardy space $H_{p}$ for $% 0<p<1/\left( 1+α\right).$
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 $0<p<1.$
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…
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 $φ(L)[0,1)$.
Extension to Walsh series of theorems of Helson and Katznelson on trigonometric series, saying that a trigonometric series whose partial sums are positive has its coefficients tend to zero but is not necessarily a Fourier-Lebesgue series
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。
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.
We prove that there exists a martingale $f\in H_p $ such that the subsequence $\{L_{2^n}f \}$ of Nörlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space $weak-L_p $ for $0<p<1 $. Moreover, we prove that for any $f\in L_p(G),$ $p\geq 1, $ $L_{2^n}f$ converge to $f$ at any Lebesgue point $x$. Some new related inequalities are derived.
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.
As main result we prove that Fejér means of Walsh-Kaczmarz-Fourier series are uniformly bounded operators from the Hardy martingale space $\ H_{p}$ to the Hardy martingale space $H_{p}$ for $ 0<p\leq 1/2.$
As main result we prove that Fejér means of Walsh-Fourier series are uniformly bounded operators from $\ H_{p}$ to $H_{p}$ $\left( 0<p\leq 1/2\right) .
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.
These informal notes briefly discuss Fourier inversion in terms of Gauss--Weierstrass kernels and summability.
We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the g…
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 $\mathbb{R}^2$.
Harmonic Analysis on the Space of M-positive Vectors
本文在由伸缩矩阵M定义的M-正向量空间上建立调和分析框架，引入Walsh函数类似物和Fourier变换，证明Poisson求和公式、Plancherel定理等，并刻画了函数及其Fourier变换均具紧支撑的Schwartz类类似物。
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.
In this note a general approach is suggested for comparison of operators. This is done by means of the Fourier transform of a measure. This approach is applied to comparison of approximation properties of various summability methods of the Fourier integrals (I) and to differential operators with constant coefficients (II).
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…
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…
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 …
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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