SOME PROBLEMS ON APPROXIMATIONS OF FUNCTIONS IN BANACH SPACES

Abstract

The work presented in this thesis is an attempt of the author to study the approximations of functions using the following techniques: 1. by trigonometric polynomials, using summability techniques (called Fourier Approximation) and 2. by Fixed-point Theorems. The whole range of the subject of study is covered in five chapters. A chapter wise summary of the thesis is as follows: 1.11 SUMMARY OF CHAPTERS CHAPTER 1: Chapter 1 is of introductory in nature. It contains various definitions of summability methods, trigonometric Fourier series, Fixed-point theorems and a review of some known results, to make the entire thesis self-contained. Further, the importance and some practical applications in modern technology (a real motivation behind the work) of these topics of approximations have also been mentioned. CHAPTER 2: In this chapter, we have proved four theorems on the approximation of functions (signals) belonging to Lip (a , p) -class using summability techniques. A signal (function) f E Lip (a, p) for p 1, 0 < a 1, if r2 n f(x + t)— f(x) Pdx }l/p = 0 ). The main Theorem of this chapter is: Let f e Lip (a ,p), and let T ) denote a lower triangular regular matrix with non-negative entries and row sums to . The matrix T has monotone rows and satisfies to —1I = (1) (1), If p > 1, 0 < a <1, and T also satisfies (n +1)max n, 0 an, r = 0(1), where r = {n / 2], then f - Tn (f ) = 0 (n-a ), where in (f) = Lan ksk(f ;x). (2) k =0 (ii). If p >1, a =1, then (2) is satisfied. (iii). If p =1, 0 < a <1, and T also satisfies (n +1)max{ano, ann} =0(1), then (2) is satisfied. Our Theorems extend the first two theorems of Chandra [27] and two theorems of Leindler [74] to a more general class of lower triangular regular matrices, which in turn generalize the results of Quade [111] on Cesaro matrix. CHAPTER 3: A function (signal) f E W(Lp, 4(t)), where 4(t) is positive increasing function oft, if {1 lip P dx = 0 (4(t)), (13 0). (f (x + t) — f(x))sin13 x We note that W , W)) coincides with the class Lip (4 (t), p) for 13 = 0. In this chapter, we prove: Let T (a „k) be an infinite regular triangular matrix with an 0, An.k Er_k a„,„Anm = 1, Vn & satisfyingrk o(k+1)1Aka„,„„ = 0(An.„),0 G r < n, then the degree of approximation of function -t'(x) , conjugate to a 27c-periodic function f(x) belonging to weighted class W(Lp,4(t)), (p 1) , is given by f(x)—in(x) = 0 (r0 +up (1/ n)), where it f (x) = J yr (t) cot (t / 2) dt, provided 4 (t) satisfies the following conditions 0 {7(t1 w(t) 1/p 4(t))P sin" t dt} = 0(1/ n), {:n I p kv(t)1/ (t))P dt} = 0(n°) uniformly in x, in which 8 is an arbitrary positive number with q (1-8) —1 > 0, where 1)-1+ crI =1, 1 p co, y(t) = yx (t) = 2-I [f (x + t) — f(x — t)], (x) are the matrix means of the conjugate Fourier series and the condition that {(t)/ t} is decreasing function in t, (3) holds. (1) Our Theorem extends a recent result of Mittal et al. [91] and a theorem of Lal and Nigam [68] for the functions of Lip (4(t),p) -class to the functions of weighted classW(Lp,4(t)),p?_1 (2) Our Theorem also generalizes a recent result of Mittal et al. [96] for the functions of W(Lp,4(t)) - class and two results of Qureshi ([116], [117]) for the functions of Lip a and Lip (a , p) -classes on Norlund (Np) -matrices. (3) Our Theorem also includes a result of Qureshi [118] for the functions of W(Lp,4(0) -class on (Np) -matrices. CHAPTER 4: In this chapter, we prove a theorem on the degree of approximation of function belonging to weighted W(Lp , (0), (p 1) -class by almost matrix means of its Fourier series. Our Theorem generalizes the results of Lal [69] and Qureshi [114]. CHAPTER 5: In this chapter, we prove four theorems on the best approximation using Fixed-Point theorems, under relaxed conditions, and derive a few well-known ([11], [12], [18], [37], [48], [119], [137], [138], [139], [148], [150], [151], [153], [156]) theorems as interesting corollaries. The results are given in locally convex Hausdorff space, Banach space and Hilbert space. The main Theorem of this chapter is: Let C be a closed star-shaped subset of X and f :C —> X a p-non-expansive map with f (C) compact such that f (a C) c C, where ac stands for boundary of C. Then f has a fixed-point. ❑