FOURIER SERIES APPROXIMATION BY LINEAR OPERATORS IN 𝑳𝒑-NORM

Abstract

In this thesis, we study the degree of approximation of functions belonging to certain function classes through trigonometric Fourier series using summability methods. We divide the thesis into six chapters. Chapter one is an introductory part of the thesis which deals with the upbringing of approximation theory, basic de nitions and some notations which are used throughout the thesis. Literature survey and the objective of the work done is also given in this chapter. Chapter two is about the approximation of 2 -periodic functions in the weighted Lipschitz class W(Lp; (t)) (p 1) by almost summability means of their Fourier series. We also obtain a result on the approximation of conjugate functions through almost matrix means of their conjugate Fourier series, which in turn improves some of the previous results. The deviation is measured in the corresponding weighted norm. We also discuss some corollaries derived from our main results. Chapter three deals with the approximation of functions by using -method of summability of conjugate Fourier series. Here we obtained a degree of approximation of the conjugate function e f, conjugate to a 2 -periodic function f in the generalized H older space H ;p (0 < 1; p 1) through Borel means of the conjugate Fourier series. Our result improves some of the previous result. In the fourth chapter, we obtain an estimate for the degree of approximation of functions belonging to the generalized Zygmund space Z! p (p 1) through product means of Fourier series, which generalizes and improves some of the previous results. The results are obtain in terms of the moduli of continuity. We also derive some corollaries from our theorems. In the fth chapter, we obtain a quantitative estimate of Young's theorem (well known in the classical Fourier analysis) by using matrix means which generalizes the i ii result obtained by Mazhar and Budaiwi [76]. In the sixth chapter, we study the degree of approximation of 2 -periodic functions of two variables, de ned on T2 := [􀀀 ; ] [􀀀 ; ] and belonging to certain Lipschitz classes, by means of almost Euler summability of their Fourier series. The degree of approximation so obtained depends on the modulus of continuity associated with the functions. We also derive some corollaries from our theorems for the functions of Zygmund classes.