Kent E. Morrison
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.
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We prove that the Fourier cosine series $\sum_{k=1}^{\infty}(-1)^{k+1}\frac{r^k\cos{kϕ}}{k+2}$ assumes its maximum value at $ϕ= 0$ for $ϕ\in [0, π)$ regardless of $r$ if $r \in (0, 1]$. This was first proved by Arias de Reyna and van de Lune. The more compact proof presented here is based on a generating function of the Chebyshev Polynomials.
Consider a collection $λ_1<...<λ_N$ of distinct positive integers and the quantities $$ M_1 = M_1(λ_1,...,λ_N) = \max_{0\le x \le 2π} |\sum_{j=1}^N \sin{λ_j x}| $$ and $$ M_2 = M_2(λ_1,...,λ_N) = - \min_{0\le x \le 2π} \sum_{j=1} \cos{λ_j x}. $$ Prompted by a discussion with G. Benke we prove that collections of frequencies $λ_j$ which have $M_1 = o(N)$ or $M_2 = o(N)$ are unstable, in the sense that one can perturb the $λ_j$ by one each and get $M_1 \ge c N$ and $M_2 \ge c N$.
We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.
We develop a method for calculating Riemann sums using Fourier analysis.
We prove an exponential integral estimate for the quadratic partial sums of multiple Fourier series on large sets that implies some new properties of Fourier series.
We construct a continuous function on the torus with almost everywhere divergence triangular sums of double Fourier series. An analogous theorem we also prove for eccentrical spherical sums.
In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral $\textbf{R}_{C}(m,n)$ in the form of integrals $\textbf{I}^{*}_{C}(\upsilon,b,c,λ,y), \textbf{J}_C (\upsilon,b,c,λ,y), \textbf{K}_{C} (\upsilon,b,c, λ,y), \textbf{I}_{C}(\upsilon,b,λ,y)$ and sol…
Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations
本研究引入带三角权重γ=sin y、关联傅里叶余弦-正弦与Kontorovich-Lebedev变换的广义卷积算子，证明其在双参数Lebesgue空间的有界性，给出加权L_p范数估计并得到相关卷积积分微分方程的L_1可解条件。
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.
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 $L_{1}$.
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.
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.
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.
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.
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.
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 $L_p, 1\le p\le\infty$.
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.
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 $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k) \cos\left(kt-\frac{βπ}{2}\right)$, $\sum\limits_{k=1}^{\infty}kψ(k)<\infty$, $ψ(k)\geq 0$, $β\in\mathbb{R}$.
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.
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。
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 $W^r_{β,1}$.
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 …
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.
Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application
本文通过 Young 型定理和 Saitoh 型不等式研究了高斯加权的 Fourier-cosine 卷积的有界性，获得了加权空间的新范数估计，并探讨了其在 Fredholm 第二类积分方程中的应用及可解性条件。
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。
Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized mult…
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 $φ(L)[0,1)$.
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 $0<p\leq 1.$
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 $\parallel f_β^ψ\parallel_{1}\leq1$.
New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications
本文提出 Fourier-cosine 与 Laplace 积分算子的新型多重卷积，建立酉性条件、逆表示及加权空间范数不等式，并应用于 Toeplitz 加 Hankel 型积分方程和 Barbashin 方程的可解性研究。
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.
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。
Certain relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The widest subspaces of the space of functions of bounded variation are indicated in which the cosine and sine Fourier transforms are integrable.
We investigate the square variation operator $V^2$ (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size $N$. We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on any ONS. This result is sharp and refines the classical Rademacher-Menshov theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\l…
For the third q-Bessel function (first introduced by F.H. Jackson, later rediscovered by W.Hahn in a special case and by H. Exton) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskii. As a specialization we get q-cosines and q-sines which admit q-analogues of the Fourier-cosine and Fourier-sine transforms. We get also a formula which is both an analogue of Gra…
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…
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_…
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…
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 …
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。
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。
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。
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。
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。
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…
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…
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…
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。
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。
Employing the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas, the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform) is investigated in L_2. Mapping and inversion properties are discussed, its relationship with the iterated Stieltjes transform is established. Various compositions with the Fourier cosine and sine transforms are obtained. The results are applied to the uniqueness and universality of the closed form solutions for certain new singular integral and integro-functional equations.
In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(θ)=\prod_{k=1}^n (1-e^{ikθ})$ and $Q_n(θ)=\prod_{k=1}^n (1+e^{ikθ})$. These products can be expressed using $\prod_{k=1}^n \sin\Big(\frac{kθ}{2}\Big)$ and $\prod_{k=1}^n \cos\Big(\frac{kθ}{2}\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$.
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimate…
In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if $S_n f$ stands for the $n$-th partial Fourier sum of $f$ and ${n_j}_j\subset \N$ is a lacunary sequence, then $S_{n_j} f$ is a.e. pointwise convergent for any $f\in L\log\log L$. In this paper we will show that $| \sup_{j} |S_{n_j}(f)| |_{1,\infty}\leq C |f|_{1} \log\log (10+\frac{|f|_{\infty}}{|f|_1})\:.$ As a direct consequence we obtain that $S_{n_j}f \rightarrow f $ a.e. for $f\in L\log\log L\log\log\log L$. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do no…
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 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…
Optimal Extrapolation Bounds for Sparse Fourier Sums
研究任意实频率上\(k\)稀疏傅里叶和的最优外推界，给出相关不等式改进前人结果。作为算法结果，提高了聚类频率恢复算法的聚类中心分辨率，还获得稀疏傅里叶特征空间的外部杠杆得分和转移界。
In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the $\ell_1$-partial sums of Fourier series. In particular, we consider the Lagrange interpolation polynomials based on the Lissajous-Chebyshev node points, the partial sums of the Fourier series generated by the anisotropically dilated rhombus, and the corresponding discrete partial sums.
The main aim of this paper is to investigate the quadratical partial sums of the two-dimensional Walsh-Fourier series.
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