On Double Summability of Double Conjugate Fourier Series

H K Nigam,Kusum Sharma

doi:10.1155/2012/104592

Abstract

For the first time, a theorem on double matrix summability of double conjugate Fourier series is established.

Highlights

The Fourier series of f x is given by f x ∼ a0 2
Conjugate to the series 1.1 is given by an cos nx − bn sin nx n1 and is known as conjugate Fourier series
It is well known that the corresponding conjugate function of 1.2 is defined as fx πf x t −f x−t dt

Summary

Introduction

Conjugate to the series 1.1 is given by an cos nx − bn sin nx n1 and is known as conjugate Fourier series. The double Fourier series of a function f x, y which is analogue for two variables of the series 1.1 , is given by f x, y. One can associate three conjugate series to the double Fourier series 1.4 in the following way: λm,n −βm,n cos mx cos ny m 1n 0 αm,n sin mx cos ny − δm,n cos mx sin ny γm,n sin mx sin ny , 1.6. ∞ n pm,n with the sequence of m, n th partial sums sm,n is said to summable by double matrix summability method or summable T, S if tm,n tends to a limit s as m → ∞ and n → ∞. Double matrix summability method T, S is assumed to be regular throughout this paper

Main Theorem

Lemmas

Proof of The Theorem