P. S. Bullen
In this paper the discussion of the effect of trigonometric series on the theory of integration is continued from an earlier paper by Gluchoff, Trigonometric series and theories of integration, Math.Mag., 67 (1994), 3--20.
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We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series.
In the present paper we consider the trigonometric series with (b,r)-general monotone and (b,r)-rest bounded variation coefficients. Necessary and sufficien conditions of L-convergence for such series are obtained in terms of the coefficients.
The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.
In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic functions.
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of H. Chen (2010).
On closed forms of some trigonometric series
本文针对参数替换引发奇点的三角级数，利用Choi-Srivastava定理推导出以Hurwitz zeta函数导数表示的闭式公式。
The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic functions are evaluated in terms of the digamma function.
In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made here.
We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically convergent everywhere to an everywhere finite integrable function $f(x)$, then the series is the Fourier series of $f(x)$. We also show that a singleton is a set of uniqueness. These results are general…
The present paper proposes a new condition to replace both the ($O$-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds.
A sufficient condition for uniform convergence of trigonometric series with p-bounded variation coefficients
本文针对p-有界变差系数的三角级数，在p>1情形下提出了一个比文献已知结果更弱的一致收敛充分条件。
Inspired by Menshov's representation theorem, we prove that there exists a sequence of frequecies such that any measurable (complex valued) function on R can be represented as a sum of almost everywhere convergent trigonometric series with these frequencies.
The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential generalization to $O$-regularly varying quasimonotonicity. An application is present in Section 3.
The generalized summation of divergent trigonometric series, namely by method of $σ_k(r,a)$-factors is considered in this paper. It is proved that such summation of Fourier series of periodical function $f(t)$ results in the convolution of this function with kernels $De(r,α,t)$; if the parameter $r$ is integer, these kernels are polynomial normalized basic $B$-splines of order $r-1$ $(r=1,2,\ldots)$. Also it is proved that the method of summation with $σ_k(r,a)$-factors is $F$-effective.
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.
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.
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…
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.
Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.
Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials
本文研究三角级数与切比雪夫多项式级数的一致有理逼近，建立三角 Hermite-Padé 多项式的存在唯一性准则并给出显式形式，同时构造例子说明 Hermite-Jacobi 逼近与 Hermite-Padé 逼近可以不同。
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…
We prove orthogonality relations for some analogs of trigonometric functions on a $q$-quadratic grid and introduce the corresponding $q$-Fourier series. We also discuss several other properties of this basic trigonometric system and the $q$-Fourier series.
We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (πn τ)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $τ$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.
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
We contribute to the method of trigonometric series for solving differential equations of certain non linear oscillators. Key Words: series power solution, trigonometric series.
Necessary and sufficient conditions are obtained on the function $M$ such that $\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in Ω\}$ is complete and minimal in $L^{p}(\mathbb{T}^{2})$ when $Ω^{c}=\{(0,0)\}$ and $Ω^{c} = 0\times\mathbb{Z}$. If $Ω^{c} = 0\times\mathbb{Z}_{0},$ $\mathbb{Z}_{0} = \mathbb{Z}\setminus\{0\}$ it is proved that the system $\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in Ω\}$ cannot be complete minimal in $L^{p}(\mathbb{T}^{2})$ for any $M\in L^{p}(\mathbb{T}^{2})$. In the case, $Ω^{c}=\{(0,0)\}$ necessary and conditions are found in terms of the one-dimensional case.
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.
In this paper we obtain degree of approximation of functions in Lp by operators associated with their Fourier series using integral modulus of continuity. These results generalize many know results and are proved under less stringent conditions on the infinite matrix.
Let $W_0(\mathbb R)$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $\sum\limits_{k=-\infty}^\infty c_k e^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $ϕ\in W_0(\mathbb R)$ such that $ϕ(k)=c_k$, $k\in\mathbb Z$. If $f\in W_0(\mathbb R)$, then the piecewise linear continuous function $\ell_f$ defined by $\ell_f(k)=f(k)$, $k\in\mathbb Z$, belongs to $W_0(\mathbb R)$ as well. Moreover, $\|\ell_f\|_{W_0}\le \|f\|_{W_0}$. S…
We proved recently math.CA/0510403 that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may be square integrable on the circle. Here we prove that it can even be infinitely differentiable, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a "new quasi-analyticity".
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 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 …
The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\i…
Norm bounds on Fourier series with polynomial spectra and constrained coefficients
本文针对谱为多项式的三角多项式，通过系数约束建立了L⁴范数上界，补充了前人成果并解决了特定猜想，确定了对应范数增长的精确阶。
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.
The Littlewood problem and non-harmonic Fourier series
针对 Littlewood 问题，本文建立非整数频率三角多项式在大区间上的 $L^1$ 范数定量估计，推广并统一了多项经典结果。
Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series
针对Erdős提出的加法组合学长期问题，通过三角级数的$L^1$估计证明：任意$n$元整数集含大小至少为$n/3+c\boldsymbol{\text{log log}}n$的无和子集，解决了该领域经典难题。
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $μ$ on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. The rate of $\wi…
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。
From Ingham to Nazarov's inequality: a survey on some trigonometric inequalities
本文综述了调和与非调和三角多项式在 L^1 和 L^2 范数下的若干不等式，重点介绍 Ingham 不等式、Littlewood 猜想的证明及 Nazarov 的非调和对应结果，并补充了小区间上的定量估计与更强条件下的下界。
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.
We construct a trigonometric series converging to zero everywhere on a subsequence, with coefficients tending to zero. We show that any such series must satisfy that the subsequence is very sparse, and that the support of the related distribution is quite large.
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…
We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_ε(\log^{9/22+ε}(N))$. This is an improvement over $O(\log^{1/2}(N))$ from the canonical ordering.
Miscellaneous summation, integration, and transformation formulas
本文利用傅里叶分析得到了多种杂项求和、积分与变换公式，涵盖特殊级数、积分融合、含狄利克雷特征的求和、超几何函数展开及反正切积分修改等主题。
In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollaczek polynomials. We can thus furnish a solution for a class of Cauchy integral equations.
It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 πn_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k \geq 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \emph{some} very slowly increasing sequences $(n_k)_{k \geq 1}$ this almost-independence property is preserved. For example, there exists $(n_k)_{k \geq 1}$ having bounded gaps such that the normalized sums $\sum \cos 2 πn_k x$ satisfy the c…
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