A problem on approximation by Fourier sums with monotone coefficients

Power-monotone Sequences and Fourier Series with Positive Coefficients

J. Németh has extended several basic theorems of R. P. Boas Jr. pertaining to Fourier series with positive coefficients from Lipschitz classes to generalized Lipschitz classes. The goal of the present work is to find the common root of known results of this type and to establish two theorems that are generalizations of Németh’s results. Our results can be considered as sample examples showing t...

On Sums of Fourier Coefficients of Cusp Forms

in case f(n) is the Fourier coefficient of a holomorphic or non-holomorphic cusp form. We shall first deal with the latter case, which is more complicated. Let as usual {λj = κj + 14} ∪ {0} be the discrete spectrum of the non-Euclidean Laplacian acting on SL(2,Z) –automorphic forms. Further let ρj(n) denote the n-th Fourier coefficient of the Maass wave form φj(z) corresponding to the eigenvalu...

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...

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...

Approximation by Polynomials with Coefficients