On Trigonometric Series with Monotonic Coefficients in $L^1_{\mu}$

Sergo A Episkoposian,Tigran M Grigorian

doi:10.5899/2011/jnaa-00075

Sergo A Episkoposian, Tigran M Grigorian

Open Access

https://doi.org/10.5899/2011/jnaa-00075

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Abstract

In this paper we consider the question of a representations of functions from weighed class L 1 [0, 2π] by series with monotonic coefficients concerning trigonometric systems .

Highlights

In this paper we consider the question of a representations of functions from weighed class L1μ[0, 2π] by series with monotonic coefficients concerning trigonometric systems
Riesz [1] proved that there exists a function f0(x) ∈ L1[0, 2π] so that its Fourier series with respect to the trigonometric system does not converge in L1[0, 2π]
There exist functions in the space L1[0, 2π] that cannot be represented by trigonometric series In [5] it is proved in the metric of L1. that there is a weighted space

Summary

Introduction

Riesz [1] proved that there exists a function f0(x) ∈ L1[0, 2π] so that its Fourier series with respect to the trigonometric system does not converge in L1[0, 2π]. There exist functions in the space L1[0, 2π] that cannot be represented by trigonometric series In [5] it is proved in the metric of L1. In this paper we consider the question of a representations of functions from weighed class L1μ, 0 < μ ≤ 1 by series with monotonic coefficients concerning trigonometric system.

Results

Conclusion