APPROXIMATION OF FUNCTIONS IN L (p 2 1) - SPACES USING SUMMABILITY TECHNIQUES

Abstract

This Thesis is devoted to approximation problems in various function spaces such as Lipschitz, Zygmund, Holder and Besov spaces by trigonometric polynomials using summability techniques. The whole range of subject is covered into eight chapters. The organization of Thesis is as follows: Chapter 1: Introduction This chapter is introductory in nature. It contains preliminaries, various definitions and required tools which are used afterwards in the Thesis. The motivation and inportance of the work in modern technology; i.e., real world applications have also been included in it. Chapter 2: Applications of Cesàro submethod to approximation of functions (signals) from Lip(n, p) class in Lu-norm This chapter is divided into three sections each having application of Cesàro submethod (CA-method) introduced by Armitage, Maddox [4]. In first section we generalize two theorems of Deer et. al. [29], by weakening the monotonicity conditions on the elements of the matrix rows, which in turn generalize the results of Leindler [67] to CA-method. In second section we extend the results of Mittal et al. [97] in view of Armitage, Maddox [4]. Our theorem also generalizes the results of the first section on Al,,-matrix to general matrix T. In third section we extend the results of Mohapatra, Szal [103] and Szal [114] using CA-method which also generalize the results of the previous two sections as well as the results of Deter et al.([29], [301), Leindler [67], Chandra [19] and Quade [120]. In each of three sections, our theorems give sharper estimates than some of the existing results. Chapter 3: Trigonometric approximation of functions in the generalized weighted Lipschitz class and their conjugates using linear operators Lal [61] has established two results on the degree of approximation of functions f in the Lip a and W(L, (L))(p > 1)-classes using a (C'. N)-sumrnability matrix. 1-lere it is worth to mention that the product summability methods are ,,,orc powerful than the individual suirimability methods and thus give an approximation for wider class of problems than the individual methods. Recently Singh et al. [135] have improved these results by removing the monotonicity condition on {p,} in the (C1 matrix. Very recently Mishra et al. [831 have obtained analogous results for an f(x), conjugate to a 27-periodic function f c W(L, (t))-class. In this chapter we compute the degree of approximation of functions in W(LP. e(t))(p > 1)-class and their conjugates using (C1 . T)-matrix so that earlier results come out as particular cases. We also discuss few corollaries of our main results. Chapter 4: Approximation of functions in the generalized Zygmund class by Hausdorif means of its Fourier series The degree of approximation of functions from various Lipschitz classes (Lip a, Lip(c, p), Lip((t), p), W(L. e(t)) etc.) has been studied by various investigators using different summa.bility methods such as Cesàro. Euler, Holder and their products. Each of the matrices involved in these methods is a Hausdorif matrix and the - product of two Hausdorif matrices is again a Ilausdorif matrix ([124], [125]). Also, multiplication of two Hausdorif matrices is commutative. Thus in view of these remarks, Rhoades ([129] - [130]), Singli, Srivastava [136] have studied the degree of approximation of functions in various Lipschitz classes using Hausdorif means. In this chapter we investigate the degree of approximation of a function in the generalized Zygmunci class Z'(p > 1) by Hausdorif means of its Fourier series. We also deduce a corollary and mention few applications of the work. Chapter 5: Degree of approximation of functions (signals) in Besov space using linear operators Besov spaces serve to generalize more elementary function spaces and are effective at measuring the smoothness properties of functions [33]. Recently Mohanty et al. [98] have obtained a theorem on the degree of approximation of functions in Besov space B(L) by choosing T to be a Niirlund (N)-mnatrix with, noll-imlcreasillg weights {p}. In this chapter we extend the result of Mohanty et al. [98] to the general - matrix T and also weaken the monotonicity conditions on the entries in matrix rows. - Chapter 6: Approximation of functions in Besov space by deferred Cesàro mean Continuing the work on Besov spaces, in this chapter we compute the error estimates of a function f in Besov space by deferred Cesàro mean [1] of partial sums of trigonometric Fourier series of f. We also deduce few corollaries of our main result for the second type delayed arithmetic mean [109] in Besov space as well as other function spaces such as Lipschitz, Holder spaces as a particular cases and compare these results with earlier known results. Chapter 7: Error estimation in approximation of functions by Fourier- Laguerre polynomials using matrix-Euler operators Various investigators have studied the degree of approximation of a function using different suinmability (Cesáro means of order : C6 , Harmonic means of order 1: i, Euler Eq , Niirlund N) means of its Fourier-Laguerre series at the point 'z: = 0 after replacing the continuity condition in SzegO theorem ([146], p.247) by a mitch lighter conditions. In this chapter we investigate the error estimation of a function - by (T. Eq)_means of its Fourier-Laguerre series at frontier point x = 0, where T is a general lower triangular regular matrix. In particular case, if T is a Cesáro matrix of order 1: Cl , then our theorem reduces to very recent result due to Krasniqi [57]. Chapter 8: Future Research Plan In this chapter we state few research problems in the direction of our work.