FOURIER APPROXIMATION IN CERTAIN FUNCTION SPACES

Abstract

We study the Fourier approximation in some selected function spaces using various summability methods in the thesis. Apart from rst chapter, each chapter corresponds to a problem on the Fourier approximation in a certain function space and its subspaces. The Chapter 1 is an introductory chapter. We introduce our problem and motivation behind it supported by a literature survey. We also set a framework for the presentation of our research work in the subsequent chapters. In Chapter 2, we study the Fourier approximation in weighted Lp-spaces using Hausdor summability methods. To obtain our results, we rst characterize the Hausdor matrices. Then using those characterizations, we estimate the rate of convergence of the Hausdor means of the Fourier series of the functions in W(Lp; ; ); p 1; 0 and (t) being a positive, non-decreasing function such that limt!0 (t) = 0: We also derive some corollaries from our main result. We study the trigonometric Fourier series approximation in the Besov spaces in Chapter 3. Our interest has been to estimate the rate of convergence of the matrix means of the Fourier series of Besov space functions in the Besov seminorm. Using estimates of the rate of convergence in Besov seminorm, we also estimate the rate of convergence in the Besov norm. Since some of the H older and Zygmund spaces are special cases of the Besov spaces, we derive the rate of convergence of the matrix means of the Fourier series of the function in the H older and Zygmund spaces from our results in the Besov spaces. In Chapter 4, we study the strong approximation of the Fourier series of the functions in generalized Zygmund spaces. Historically, the strong approximation techniques have been used to analyze the convergence properties of the Fourier series of continuous functions. We extend the strong approximation framework to study the convergence properties of the Fourier series of the functions in generalized Zygmund spaces, the subspaces of Lp[0; 2 ]: Chapter 5 is devoted to the study of the convergence properties of the corrected Fourier series. The corrected Fourier series was presented to overcome the Gibbs phenomena exhibited by the Fourier series. It is a combination of the trigonometric Fourier series, algebraic polynomials and Heaviside functions. We estimate the rate of convergence of the matrix means of the corrected Fourier series. We also demonstrate some examples to highlight the advantages, the corrected Fourier series o ers. With the introduction of inner product spaces, an abstract de nition of the Fourier series was presented. It gave rise to various Fourier series other than trigonometric. One of them is the Mellin - Fourier series. The Mellin - Fourier series is de ned for the recurrent and c-recurrent functions. There are some convergence studies of the Mellin - Fourier series available in the literature. Extending the prior convergence studies, in Chapter 6, we study the convergence properties of the matrix means of the Mellin - Fourier series.