Integrability and L1-convergence of trigonometric series with real and complex coefficients

Abstract

The present thesis entitled \lq \lq \textbf{Integrability and $L^1$-convergence of trigonometric series with real and complex coefficients}" comprises certain investigations carried out by me at the School of Mathematics, Thapar Institute of Engineering and Technology, Patiala (TIET), under the supervision of Dr. Jatinderdeep Kaur, Assistant Professor, School of Mathematics, TIET, Patiala and Dr . S.S. Bhatia, Professor, School of Mathematics, TIET, Patiala.

The existence of sine and cosine series as a Fourier series, their integrability and $L^1$-convergence seems to be one of the difficult question in theory of convergence of trigonometric series in $L^1$-metric norm. It is well known that if a trigonometric series converges in $L^1$-metric to a function $f \in L^1(T)$, then it is a Fourier series of the function $f$ but the converse of the said result does not hold good. The present thesis has been written on the same basis. In this thesis, Integrability and $L^1$-convergence of trigonometric cosine and sine series have been studied by imposing different conditions on coefficient sequences as well as by introducing new modified trigonometric cosine and sine sums \textit{as these sums approximate their limit better than the classical trigonometric series in the sense that they converge in $L^1$-metric to the sum of trigonometric series whereas the classical series itself may not.}

The thesis embodies six chapters. \textbf{Chapter I,} is introductory. In this chapter, apart from setting up the notations and terminologies to be used in the sequel, some known results interrelated with the work done in the present thesis have been presented. Further a brief plan of the results presented in the subsequent chapters is given in this chapter.

In \textbf{Chapter II,} a new modified trigonometric cosine sum has been introduced and criterion for the summability and $L^1$-convergence of modified cosine sum has been obtained under the class of generalized semi-convex sequences. Also, the $L^1$-convergence of the cosine series has been deduced as corollary in this chapter.

In the literature so far available, most of the authors have studied the integrability and $L^1$-convergence of trigonometric cosine series using different classes of coefficient sequences. However, very few of them have studied the $L^1$-convergence of the trigonometric sine series. In \textbf{Chapter III,} a new modified trigonometric sine sum has been introduced and its $L^1$-convergence has been studied under the class $\tilde{C}\bigcap \tilde{B}V$. Further, in this chapter we have obtained $L^1$-convergence of $r^{th}$ derivative of modified trigonometric sine sum under a new extended class $\tilde{C}_r \bigcap \tilde{B}V_r ~(r=0,1,2,...)$ of coefficient sequences. Also, an application has been given to illustrate the main result.

\textbf{Chapter IV,} is devoted to the study of $L^1$-convergence of complex form of the new modified trigonometric cosine and sine sums introduced in chapter II and III under the new generalized classes $\mathbb{JS}^{c}$ and $\mathbb{JS}_2^{c}$ of complex coefficient sequences.

In \textbf{Chapter V,} we have introduced a new modified double trigonometric sine sum and have studied the integrability and $L^1$-convergence of double trigonometric cosine series under the new class $\mathcal{S}_{jk}$ of double coefficient sequences.

The objective of \textbf{Chapter VI} is to study the integrability and $L^1$-convergence of Fourier series of periodic fuzzy valued functions with new class $\mathbf{K_{p_1}(F)}$ of fuzzy coefficients.