Generalized Localization for Spherical Partial Sums of Multiple Fourier Series

Exponential Approximations Using Fourier Series Partial Sums

The problem of accurately reconstructing a piece-wise smooth, 2π-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f an...

The Approximation by Partial Sums of Fourier Series

uniformly in x as h—»0, we shall say briefly that/ belongs to Lip a. Following 4](1)) we shall say that/ belongs to a) uniformly as *—>0. a notation already used (see Zygmund lipa, 0<a<l, if \f(x+h)-f(x)\=o(\h It is a classical result of Lebesgue (see Zygmund [5, p. 61 ] ; hereafter this book will be denoted by T.S.) that if / belongs to Lip a and if sn denotes the nth partial sum of the Fourie...

Rational Trigonometric Approximations Using Fourier Series Partial Sums

A class of approximation.s {,5'N,M } to a periodic function f which uses tile ideas of Pad6, or rational function, approximations based on the Fourier series representation of f, rather than oil the Taylor series representation of f, is introduced and studied. Each approximation ,q'X,M is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coef...

Summation of Multiple Fourier Series by Spherical Means.

A Note on the Fourier Coefficients and Partial Sums of Vilenkin-fourier Series

The aim of this paper is to investigate Paley type and HardyLittlewood type inequalities and strong convergence theorem of partial sums of Vilenkin-Fourier series. Let N+ denote the set of the positive integers, N := N+ ∪ {0}. Let m := (m0,m1, . . .) denote a sequence of the positive numbers, not less than 2. Denote by Zmk := {0, 1, . . . ,mk − 1} the additive group of integers modulomk. Define...