A new condition for the uniform convergence of certain trigonometric series

Uniform approximation of periodical functions by trigonometric sums of a special type

The approximation properties of the trigonometric sums U_{n,p}^ψof a special type are investigated on the classes C^ψ_{β, \infty} of (ψ,β)-differentiable (in the sense of Stepanets) periodical functions. The solution of Kolmogorov-Nikol'skii problem in a sufficiently general case is found as a result of consistency between the parameters of approximating sums and approximated classes. It is shown that, in some important cases the sums under consideration provide higher order of approximation in the uniform metric on the classes C^ψ_{β, \infty} than Fourier sums, Zygmund sums and de la Valle Po…

Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

Chaundy and Jolliffe [4] proved that if $\{a_{n}\}$ is a non-increasing (monotonic) real sequence with $\lim\limits_{n\to \infty}a_{n}=0$, then a necessary and sufficient condition for the uniform convergence of the series $\sum_{n=1}^{\infty}a_{n}\sin nx$ is $ \lim\limits_{n\to \infty}na_{n}=0$. We generalize (or weaken) the monotonic condition on the coefficient sequence $\{a_{n}\}$ in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy and J…

A remark on the trigonometric system

We present a new property of the trigonometric system arranged in a natural order. It is shown that the sequence of subspaces of trigonometric polynomials is optimal in the sense of order of convergence on convolution classes K*Up in Lq for any 1<p,q<\infty just in the cases of "very slow" or "very fast" rate of decay of Fourier coefficients of K.

Rearrangements of Trigonometric Series and Trigonometric Polynomials

The paper is related to the following question of P.~L.~Ul'yanov: is it true that for any $2π$-periodic continuous function $f$ there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of $f$ decrease. Also, we study a problem how to choose $m$ terms of a trigonometric polynomial of degree $n$ to make the uniform norm of their sum as small as possible.

Summability of Multi-Dimensional Trigonometric Fourier Series

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…

Trigonometric series with noninteger harmonics

Let $\{c_k\}$ be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and $α>0$ be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series $\sum_k c_k\sin k^αx$ and $\sum_k c_k\cos k^αx$ on the real line and its bounded subsets.

On Weyl multipliers of the rearranged trigonometric system

We prove that the condition \begin{equation} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation} is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property for Haar, Walsh, Franklin and some other classical orthogonal systems was known long ago. The proof of this result is based on a new sharp logarithmic lower bound on $L^2$ of the majorant operator related to the rearranged trigonometric system.

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。

On the best approximation of certain classes of periodic functions by trigonometric polynomials

We obtain the estimates for the best approximation in the uniform metric of the classes of $2π$-periodic functions whose $(ψ,β)$-derivatives have a given majorant $ω$ of the modulus of continuity. It is shown that the estimates obtained here are asymptotically exact under some natural conditions on the parameters $ψ,$ $ω$ and $β$ defining the classes

On L1 Convergence of Fourier Series of Complex Valued Functions

In the present paper, we give a brief review of $L^{1}$-convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.

Strong boundedness, strong convergence and generalized variation

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $L^{p}$ spaces, $1\leq p < \infty$. We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.

缺项三角和的中偏差原理与Mod-Gaussian收敛

Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums

本文研究缺项三角和的中偏差原理与Mod-Gaussian收敛，引入相关图并采用累积量方法，证明√n与n/log(n)尺度间无算术效应，还借助不同工具得到mod-Gaussian收敛结果，为明确大偏差特殊现象的显现尺度提供研究基础。

Expansion of real valued meromorphic functions into Fourier trigonometric series

In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of improper integrals convergence of parametric meromorphic functions is defined. In accordance with this, it is shown that the …

On parametric Thue-Morse Sequences and Lacunary Trigonometric Products

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n α)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $α$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k α)_{k \geq 1}$ is u.d. mod 1 for almost all $α$. However, in this general case it is usually extremely difficult to classify those $α$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical …

双曲交叉三角多项式一致范数的采样离散化

Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials

研究双曲交叉三角多项式一致范数的采样离散化，构造显式赋范集并证明随机采样达到最优指数，同时给出赋范集大小的下界，排除二次界。

On almost everywhere convergence of Malmquist-Takenaka series

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of $z$ are replaced by products of other Möbius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove $L^p$ bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the Möbius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theo…

Bohr--Favard Inequality for differences and constants in Jackson--Stechkin Theorem

We consider uniform approximations by trigonometric polynomials. The aim of the paper is to obtain good estimates of the Jackson--Stechkin constants $J_m$. We prove that $ J_m \le C 2^{-m+5/2\log_2m}$. Our proof is based on the difference analogue of the Bohr--Favard inequality.

对 Fejér 和 Hardy 古典定理的新视角

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。

Uniform convergence of hypergeometric series

The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions can be checked algorithmically. Hence the results can be applied in Zeilberger type algorithms for nonterminating hypergeometric series.

Weighted uniform convergence of entire Grünwald operators on the real line

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type $τ>0$ at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

Uniform convergence of Fourier-Bessel series on a q-linear grid

We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier-Bessel series.

A new class of numerical sequences and its applications to uniform convergence of sine series

In the present paper we introduce a new class of sequences called GM(b,r), which is the generalization of a class considered by Tikhonov. Moreover, we obtained in this note sufficient and necessary conditions for uniform convergence of sine series with (b,r)-general monotone coefficients.

On the uniform convergence of double sine series

The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [3], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are …

On uniform convergence of Fourier series

We consider the space $U(\mathbb T)$ of all continuous functions on the circle $\mathbb T$ with uniformly convergent Fourier series. We show that if $φ: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but not linear map, then $\|e^{inφ}\|_{U(\mathbb T)}\simeq\log n$.

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…

Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one dimensional torus translation, these estimates are nearly optimal.

Uniform convergence criterion for non-harmonic sine series

We show that for a nonnegative monotone sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for uniform convergence of the series $\sum_{k=1}^{\infty}c_k\sin k^α x$ on any bounded set for $α\in (0,2)$, and for an odd natural $α$ it is sufficient for uniform convergence on the whole $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^α$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of an even $α$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for convergence of the mentioned series at the point $π/2$ or at the point…

A Simple Proof of the Uniform Convergence of Fourier Series in Solutions to the Wave Equation

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.

二次欧几里得空间中熵势的统一收敛速率

A uniform rate of convergence for the entropic potentials in the quadratic Euclidean setting

本文研究了二次欧几里得空间中熵势及其梯度向Brenier势和其梯度的统一收敛速率，适用于满足凸性假设的绝对连续测度。