A Study Of Summability Of Derived ourier Series And Its Conjugate By Matrix Method

Abstract

There are many types of criteria, under various conditions, for the matrix summation of a derivative Fourier series. Here, we will study an important and distinct type of criteria for the matrix summation of a derivative Fourier series. We will consider the function f(x) to be periodic with a period of 2π and Liebig integrable on the interval [-π, π], and the matrix T = (a_{n,k})$ to be regular. We will prove that the series \sum_{n=1}^{\infty}nB_{n}(x) is matrix summation to the sum $O(1), which is a bounded function, as we will see later, and that the series -\sum_{n=1}^{\infty}nA_{n}(x) is matrix summation to the sum: 14

-\frac{1}{4\pi}\int_{0}^{\pi}h(t).cosec^{2}\frac{1}{2}tdt+O(1)