Mean-Square Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space

Mirgand Sh Shabozov,Mukim S Saidusaynov

doi:10.18052/www.scipress.com/ijarm.9.34

Abstract

The problem of mean-square approximation of complex variables functions regularly in some simply connected domainDc C with Fourier series by orthogonal system when the weighted functionγ: =γ(|z|) is nonnegative integrable inD, was considered. An exact convergence rate of Fourier series by orthogonal system of functions on some class of functions given by special module of continuity ofm-thorder were obtained. An exact values ofn-widthsfor specified class of functions were calculated.

Highlights

Introduction and Preliminary ResultsIn this paper, the quadratic approximation of functions with Fourier series by orthogonal system over complex variable domain in the presence of weight was considered.In the domain of D ⊂ C is given a nonnegative measurable and not equivalent zero function γ(|z|), such that there is exists a finite integral∫∫ ‍γ(|z|)dσ > 0, (D)where the integral is understood in sense of Lebesgue and dσ the element of area
Where the integral is understood in sense of Lebesgue and dσ the element of area
We will consider the problems of mean square approximation by Fourier sum of complex function f regularly in the connected domain D and is belong to the space L2,γ: = L2(γ(|z|), D) with finite norm

Summary

Introduction

Introduction and Preliminary ResultsIn this paper, the quadratic approximation of functions with Fourier series by orthogonal system over complex variable domain in the presence of weight was considered.In the domain of D ⊂ C is given a nonnegative measurable and not equivalent zero function γ(|z|), such that there is exists a finite integral∫∫ ‍γ(|z|)dσ > 0, (D)where the integral is understood in sense of Lebesgue and dσ the element of area. Let {φk(z)}k∞=0 be complete orthonormal system in domain D of a system of complex functions in the space L2,γ: Using the generalized translation operator Fh(f) for an arbitrary function f ∈ L2,γ, we define the finite-difference of first and higher order by the equations In work [9] it was proved that for any arbitrary function f ∈ L2,γ for each t ∈ (0,1) is hold an estimate

Results

Conclusion