L1 -Convergence of Newly Defined Trigonometric Sums Under Some New Class of Fourier Coefficients

Abstract

Tough difficulties in the trigonometric series convergence in L1 norm is appearance of trigonometric series as Fourier series, and its L
1- convergence. Many academics investigated trigonometric series separately by examining the cosine & sine series , so as a result, modified cosine sums and sine sums were developed to assess the sharp consequences on trigonometric series’s integrability & L1
-convergence, as improved sums approach respective limits closer than traditional trigonometric sums. This work presents ‘KP’, a new class of Fourier Coefficients, as well as advanced cosine and sine sums of trigonometric series with real coefficients. As a result, necessary & sufficient criterion for Integrability and L1-normed convergence for trigonometric functions is achieved. Here, authors also discuss about L1-convergence of r the differential of newly defined improved trigonometric sums with Fourier coefficients are from an enlarged class KPr