L^1-convergence of complex double Fourier series

On the Convergence of Random Fourier--Jacobi Series in $L_{[-1,1]}^{p,(η,τ)}$ space

Liu and Liu introduced the random Fourier transform, which is a random Fourier series in Hermite functions, and applied it to image encryption and decryption. They expected its applications in optics and information technology. These motivated us to look into random Fourier series in orthogonal polynomials. Recently, we have established the convergence of random Fourier--Jacobi series $\sum\limits_{n=0}^\infty d_n r_n(ω)φ_n(y),$ where $φ_n(y)$ are the orthonormal Jacobi polynomials $p_n^{(γ,δ)}(y),$ $r_n(ω)$ are random variables associated with stochastic processes like the Wiener process, the…

连续函数的随机Fourier-Jacobi级数的收敛性

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

本文研究Jacobi多项式随机Fourier级数的随机收敛性，针对不同参数和连续函数系数建立收敛结果，并指出和函数为随机积分且讨论其连续性。

A Result on Convergence of Double Sequences of Measurable Functions

We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.

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.

On absolute convergence of Fourier integrals

New sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension $d\ge2.$

Mean convergence of Fourier-Dunkl series

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the $A_p$ classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.

Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces

In this paper, we have defined rough convergence and rough statistical convergence of double sequences in probabilistic normed spaces which is more generalized version than the rough statistical convergence of double sequences in normed linear spaces. Also, we have defined rough statistical cluster points of double sequences and then, investigated some important results associated with the set of rough statistical limits of double sequences in these spaces. Moreover, in the same spaces, we have proved an important relation between the set of all rough statistical cluster points and rough stati…

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 $0<p<1.$

离散群上的傅里叶重排：$L^p$ 与算子范数子序列收敛

Fourier rearrangements on discrete groups: $L^p$ and operator-norm subsequence convergence

本文针对双曲群或多项式增长群，构造傅里叶项重排使子序列在算子范数及$L^p$范数下收敛，证实Révész性质与多项式增长猜想，否证自由群反例猜想。

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.

An Operator theoretic approach to the convergence of rearranged Fourier series

This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on $B(L_2(T))$. Additionally, a new framework for further investigation is introduced by consideri…

Rough I-statistical convergence of double sequence

The concept of I-statistical convergence of a double sequence was first introduced and study by Das et. el [2]. Here in this paper we discuss some results on rough ideal statistical convergence and also we introduce the notion of rough ideal limit set and discuss some topological aspects on this set.

一般傅里叶级数的无条件收敛

Unconditional convergence of general Fourier series

本文研究一般正交规范系统下Lip_1函数傅里叶级数的无条件收敛条件，证明了最优条件的存在性，并发现任何ONS均含使无条件收敛成立的子系统。

Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems

We construct a simple example of an integrable function on the ring of integers of the $p$-adic field $\Q_p$ having an almost everywhere divergent Fourier series. On the other hand, we prove the pointwise convergence of the Fourier series of functions in $L^p(\D,w)$, $1<p<\infty$, where $\D$ is the ring of integers of a local field $K$ and $w$ is a weight in the Muckenhoupt $A_p$ class. This result includes, as special cases, when $\D$ is the ring of integers of $\Q_p$ or the field $\mathbb{F}_q((X))$ of formal Laurent series over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is …

Approximation properties of the double Fourier sphere method

We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical Hölder spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the res…

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable gr…

$d$维流形的双重傅里叶球面方法

A double Fourier sphere method for $d$-dimensional manifolds

本文提出一种广义双重傅里叶球面（DFS）方法，将定义在$d$维流形上的函数变换到同维环面进行傅里叶逼近，并证明其一致收敛性及Hölder连续函数的解析收敛速率。

Almost-everywhere convergence of Fourier series for functions in Sobolev spaces

Let $S_λF(x)$ be the spherical partial sums of the multiple Fourier series of function $F\in L_2(\mathbb{T}^N)$. We prove almost-everywhere convergence $S_λF(x)\rightarrow F(x)$ for functions in Sobolev spaces $H_p^a(\mathbb{T}^N)$ provided $1< p \leq 2$ and $a> (N-1)(\frac{1}{p}-\frac{1}{2})$. For multiple Fourier integrals this is well known result of Carbery and Soria (1988). To prove our result, we first extend the transplantation technic of Kenig and Tomas (1980) from $L_p$ spaces to $H_p^a$ spaces, then apply it to the Carbery and Soria result.

Multiple Fourier series and lattice point problems

For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series, involving the convergence properties of them. The third phenomenon is closely related to the lattice point problems, which is a classical theme of the analytic number theory. We also prove that, for the case of two or three dimension, the convergence problem on the Fourier series is equivalent to the lattice point problems …

三角形区域上 Fourier 级数的 $L^p$ 收敛性

The $L^p$ convergence of Fourier series on triangular domains

本文针对 45-90-45、等边和半等边三类三角形，证明 Dirichlet Laplacian 特征函数 Fourier 级数适当截断的 $L^p$ 范数收敛，并依据 Lamé 定理说明方法局限。

A note on rough $I$-convergence of double sequences

In this paper we study some basic properties of rough $I$-convergent double sequences in the line of D$\ddot{u}$ndar [8]. We also study the set of all rough $I$-limits of a double sequence and relation between boundedness and rough $I$-convergence of a double sequence.

Pointwise convergence of noncommutative Fourier series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fouri…

Fourier--Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier--Jacobi expansions in Morrey spaces.

关于涉及非紧算子的Fourier谱方法收敛性的研究

On the convergence of Fourier spectral methods involving non-compact operators

受Fredholm理论启发，本文提出了一个建立算子方程谱方法收敛性的框架，并应用于Fourier有限截断和配置法求解周期边界微分方程及Riemann--Hilbert问题，改进了周期系数微分算子特征值逼近结果。

A note on the Fourier coefficients and partial sums of Vilenkin-Fourier series

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.

Norm resolvent convergence of discretized Fourier multipliers

We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff di…

关于圆盘上 Bessel–Fourier 级数的 $L^p$ 收敛性的评述

Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc

本文评述圆盘上 Bessel–Fourier 级数的收敛问题，概述 Balodis–Córdoba 的混合范数结果，并将其修改推广到 $2\leq p<4$ 时的 $L^p(\mathbb{D},r\mathrm{d}r\mathrm{d}t)$ 范数收敛。