A Study Of Summability Of Derived ourier Series And Its Conjugate By Matrix Method - Amer, Mohamed Mahmoud (Journal of natural sciences, life and applied sciences, 2019)
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}^{∞}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}^{∞}nA_{n}(x) is matrix summation to the sum: 14-\frac{1}{4π}\int_{0}^{π}h(t).cosec^{2}\frac{1}{2}tdt+O(1)
Keywords: Derived Fourier Series; Conjugate Of Derived Fourier Series; Toeplitz Matrix; Absolutely Continuous Matrix Summability; Riemman Lebesgue Theorem; Matrix Hankel; Zygmund