On L1-Convergence of Trigonometric Sine Series with Special Coefficients

Abstract

The present dissertation entitled \L1􀀀Convergence of trigonometric sine series with special coe cients"contains a brief account of study carried out by me on L1􀀀convergence of Trigonometric Sine Sums under the supervision of Dr. Jatin- derdeep Kaur, Assistant Professor, School of Mathematics, Thapar Institute of Engineering and Technology, Patiala. In the literature so far available, very few work has been done concerning the con- vergence of trigonometric sine series in L1􀀀norm. Keeping this in view, The L1􀀀 convergence of trigonometric sine series under di erent conditions on coe cients have been studied. Also, several authors introduced modi ed trigonometric cosine and sine sums. The rst chapter is introductory. In this chapter, apart from setting up the notations and terminology to be used in sequel, we have presented some known results. The purpose of chapter II is to study the L1􀀀convergence of trigonometric sine series us- ing class eS1. In the chapter III, the result on L1􀀀convergence of trigonometric sine series has been obtained using Ram and Kumari Modi ed sine sums

Xn k=1 Xn j=k 4

aj j

k sin kx ! un- der the classes e BV T e C. In chapter IV, L1􀀀convergence of r times di erentiable trigonometric sine series has been studied using the classes e BVr(r = 0; 1; 2; 3; : : : ) and e Cr(r = 0; 1; 2; 3; : : : ). In the end, references of various publications cited in the present dissertation have been reported. 1