An Application of Fourier Analysis to Riemann Sums

A Generalization of Riemann Sums

We generalize the property that Riemann sums of a continuous function corresponding to equidistant subdivision of an interval converge to the integral of that function, and we give some applications of this generalization.

Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the Möbius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the Möbius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

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.

The error for the second moment of cotangent sums related to the Riemann Hypothesis

In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error term in these asymptotic results. In the present paper the authors establish a lower bound for the error term for the second moment.

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 $L_p, 1\le p\le\infty$.

On Riemann sums and maximal functions in $\ZR^n$

In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.

Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound - essentially the number of terms - by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

关于左和右黎曼和的单调性

On the monotonicity of left and right Riemann sums

本文研究了左和右黎曼和的单调性，回顾了相关文献，提出了新的充分必要条件，并利用主导理论给出了基本定理的新证明，还深入探讨了Borwein提出的问题。

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 …

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 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.

A problem concerning Riemann sums

An open problem concerning Riemann sums, posed by O. Furdui, is considered.

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 $\parallel f_β^ψ\parallel_{1}\leq1$.

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.

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions

We consider two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$, where $ζ(a,x)$ is the Hurwitz zeta function, given by $$\int_0^1ζ_1(a,x)ζ_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1ζ_1(a,x)ζ_1(b,1-x)\,dx$$ when $\Re (a,b)>1$. These integrals have been investigated recently in \cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $ζ(x)$ and two free parameters $a$ and $b$. The limiting forms of…

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。

On the maximum of cotangent sums related to the Riemann Hypothesis in rational numbers in short intervals

Cotangent sums play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

On Power Sums of Positive Numbers

In this work we establish a necessary and sufficient condition for a genus $0$ entire function $f(z)$ has only positive zeros by applying Hausdorff moment problem and Mergelyan's theorem, the obtained criterion is very much reminiscent of Xian-Jin Li's criterion on the Riemann hypothesis. We also apply this criterion to the Riemann hypothesis and the generalized Riemann hypothesis for certain Dirichlet $L$-series of real primitive characters.

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.

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…

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.

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。

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 $φ(L)[0,1)$.

On the partial sums of Vilenkin-Fourier series

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 $0<p\leq 1.$

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…

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…

稀疏均值估计、代数数解计数与非阿基米德傅里叶分析

Sparse mean value estimates, algebraic number solution counting, and non-Archimedean Fourier analysis

本文利用非阿基米德局部域上的傅里叶解耦定理，应用于实变量问题，得到指数和均值估计与代数数解计数界，并指出实与p-进解耦估计的差异。

累积黎曼和、分布函数与一个通用不等式

Cumulative Riemann sums, distribution functions, and a universal inequality

本文研究了累积黎曼和与分布函数的关系，推导了一个通用不等式，并探讨了其与主次理论和卡拉马塔不等式的关系。

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。

Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods

In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials $p_{n-1}(z) = \sum_{k=0}^{n-1} z^k/ k!$. Our proof is based on a representation of $p_{n-1}(nz)$ in terms of an integral of the form $\int_γ \frac{e^{nϕ(s)}}{s-z}ds$. We demonstrate how to derive uniform expansions for such integrals using a Riemann-Hilbert approach. A comparison with classical steepest des…

Bessel函数级数作为Riemann zeta函数级数的表示

Series over Bessel functions as series in terms of Riemann's zeta function

本文利用Hurwitz公式和三角级数求和公式，将Bessel函数级数转化为Riemann zeta函数级数，并求得有限和。

Partial sums of the M{ö}bius function

Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates of zeros of the Riemann zeta-function.

Riemannian流形上的Fourier不确定性原理

Fourier Uncertainty Principles on Riemannian Manifolds

本文研究紧致Riemannian流形上的Fourier不确定性原理，指出特征函数增长是关键障碍，并探讨了其与局部紧Abel群背景及Bourgain的$Λ_q$定理的联系。

Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a …

An Euler-Maclaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

On discrete fractional integral operators and mean values of Weyl Sums

In this paper we prove new $(\ell^p, \ell^q)$ bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number $r_{s,k}(l)$ of representations of a positive integer $l$ as a sum of $s$ positive $k$-th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a fur…

Fourier transforms of positive definite kernels and the Riemann $ξ$-Function

The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between the Bochner-Khinchin-Mathias theory of positive definite kernels and the generalized real Laguerre inequalities. The concavity and convexity properties of the Jacobi theta function play a prominent role throughout this work. The paper concludes with several questions and open problems.