Pointwise Fourier inversion on tori and other compact manifolds

We prove a number of results about pointwise convergence of eigenfunction expansions of functions on compact manifolds. In particular, we establish that the Pinsky phenomenon holds for piecewise smooth functions on the three-dimensional torus, with jump across the boundary of a ball, in the same form as it was discovered for functions on three-dimensional Euclidean space. Our work on this has been stimulated by recent work of Brandolini and Colzani, and we also discuss some variants of their results.     

Pointwise Fourier inversion of distributions

We show that,given a tempered distribution T whose Fourier transform is a function of polynomial growth and a point x in Rn at which T has the value τ(in the sense of Lojasiewicz),the Fourier integral of T at x is summable in Bochner-Riesz means to τ.     

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.     

关于Taylor级数与Fourier级数的几点注记

Taylor级数与Fourier级数是两类非常重要的函数项级数,二者在发展与应用背景、展开条件、收敛性和展开的唯一性等方面不尽相同,本文对此作了一些总结与探讨。     

Periodic hyperfunctions and Fourier series

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients of periodic hyperfunctions are of infra-exponential growth in , i.e., for every and every . This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions.

To show these we introduce the space of hyperfunctions of growth which generalizes the space of distributions of growth and represent generalized functions as the initial values of smooth solutions of the heat equation.

     

Optimal control of linear time-varying systems via Fourier series

The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.     

On vector-valued random Fourier series

Consider a (complex) Banach spaceX, such thatX C_O, and vectors(X _i ) _i ofX. Consider an independent standard normal sequence(g _i ) _i . Then if anX-valued random Fourier series _{|k| n} e ^ikt g _k x _k satisfies

Fourier级数两个收敛定理的关系及反例

Fourier级数的收敛问题一直是很多数学研究者关注的问题,不同的数学分析教材和高等数学教材对收敛定理的表述各不相同。本文通过实例说明这些收敛定理之间不存在包含关系,并进一步说明这些收敛定理都是判别Fourier级数收敛的充分条件,而不是充要条件。     

Generalized absolute Cesaro summability of a Fourier series

In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK|t∈B _u V in (0,π)⟹ΣA _n (x) is |C, 0,β| forβ>1, at the pointt = x.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set there exists a homeomorphism of the closed interval such that for an arbitrary function f the Fourier series of the function F(x,y) = f((x),(y)) converges uniformly on simultaneously over rectangles, over spheres, and over triangles.     

Fourier series and the Lubkin W-transform

We discuss the effect of a particular sequence acceleration method, the Lubkin W-transform, on the partial sums of Fourier series. We consider a very general class of functions with a single jump discontinuity, and prove that this method fails on a large set of points. The authors were partially supported by the Kansas State University REU and NSF grant GOMT530725.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Fourier coefficients of Siegel-Eisenstein series of odd genus

We give a new expression of the T-th Fourier coefficients of the Siegel-Eisenstein series of odd genus. Unlike even genus case, only modified divisor sums and no generalized class numbers enter into our formula.     

两种基本函数项级数系数公式注记

分析部分高等数学教材在推导幂级数和傅里叶级数的系数公式时存在的不足,并从函数逼近论的角度,就此问题提出改进思想和方法.     

利用函数的傅里叶展开式求级数的和

利用函数的傅里叶展开式可求得级数∞∑n=11/n2+λ2及∞∑n=1(-1)m/n2+λ2的和,而通过引入复数并利用欧拉公式可求得级数∞∑n=1 1/n2+λ2及∞∑n=1(-1)m/n2+λ2的和.     

关于Fourier级数的一点推广

以２ｌ为周期的函数ｆ（ｘ）也可看作周期为４ｌ．设ｆ（ｘ）满足 Ｄｉｒｉｃｈｌｅｔ充分条件,按［１］方法展开的以 ２ｌ为周期的 Ｆｏｕｒｉｅｒ 级数和以 ４ｌ为周期的 Ｆｏｕｒｉｅｒ 级数则对应于不同表达式．本文证明了这两种表达形式是一致的     

Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points

The existence of the `rare' sequence of partial sums summable with the method of arithmetical means at each Lebesgue point is proved in the paper. The proof is based on the strategy of random choice.

     

Integrability Properties of the Maximal Operator on Partial Sums of Fourier Series in Orlicz Spaces

Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a₀ < 1 such that then we have .