FOURIER SERIES APPROXIMATION OF FUNCTIONS IN Lp (p>1) - SPACES

Abstract

In the present thesis, we study the degree of approximation of functions belonging to certain Lipschitz classes and subclasses of Lp(p 1)-space (written as Lp throughout the thesis) through trigonometric Fourier series. We also study the approximation properties of some functions by means of Walsh{Fourier series and Fourier{Laguerre series. This thesis is divided into six chapters. Chapter 1 is introductory in nature and includes the introduction of di erent Fourier series, basic de nitions, concepts and the literature review. The objective of the work done and layout of the thesis is also given in this chapter. In Chapter 2, we estimate the pointwise approximation of periodic functions belonging to Lp(!) (or Lp(~!) )-class, where !(or ~!) is an integral modulus of continuity type function associated with f, and its conjugate function using product summability generated by the product of two general linear operators. We also measure the degree of approximation in the weighted norm for a function f belonging to weighted Lipschitz class W(Lp; (t)) and its conjugate ~ f, respectively. We prove the following theorems in this chapter: Theorem 2.3.1 Let f 2 Lp(!) with 0 < < 1 􀀀 1 p ; p > 1; and the entries of the lower triangular matrices A (an;k) and B (bn;k) satisfy the following conditions: bn;n 1 n + 1 ; n 2 N0; (0.0.1) jbn;mam;0 􀀀 bn;m+1am+1;1j bn;m (m + 1)2 for 0 m n 􀀀 1 (0.0.2) and mX􀀀1 k=0 j(bn;mam;m􀀀k 􀀀 bn;m+1am+1;m+1􀀀k) 􀀀 (bn;mam;m􀀀k􀀀1 􀀀 bn;m+1am+1;m􀀀k)j i ii

bn;m (m + 1)2 for 0 m n 􀀀 1; (0.0.3) with An;n = Bn;n = 1 for n = 0; 1; 2; ::: . Then the degree of approximation of f by BA means of its Fourier series is given by

tBA n (f; x) 􀀀 f(x)

= Ox

Xn m=0 bn;m m + 1 (n + 1) +1!(1=(n + 1)) ! ; provided that the positive non-decreasing function ! satis es the following conditions: !(t)=t is a non-increasing function; (0.0.4) (Z =(n+1) 0

j (x; t)j sin (t=2) !(t) p dt )1=p = Ox 􀀀 (n + 1)􀀀1=p ; (0.0.5) Z =(n+1)

t􀀀 j (x; t)j sin (t=2) !(t) p dt 1=p = Ox 􀀀 (n + 1) 􀀀1=p ; (0.0.6) where is an arbitrary number such that 1=p < < + 1=p and p􀀀1 + q􀀀1 = 1: Theorem 2.3.2 Let f be a 2 -periodic function belonging to the class Lp(~!) ; 0 < < 1=p; p > 1 and the entries of the lower triangular matrices A (an;k) and B (bn;k) satisfy the following conditions: bn;n 1 n + 1 ; n 2 N0; (0.0.7) jbn;mam;m􀀀l 􀀀 bn;m+1am+1;m+1􀀀lj bn;m (m + 1)2 for 0 l m n 􀀀 1: (0.0.8) with An;n = Bn;n = 1 for n = 0; 1; 2; ::: . Then the degree of approximation of ~ f; conjugate of f, by BA means of its conjugate Fourier series is given by

~tBA n (f; x) 􀀀 ~ f(x)

= Ox

Xn m=0 bn;m m + 1 (n + 1) +1~!(1=(n + 1)) ! ; provided that the positive non-decreasing function ~! satis es the following conditions: ~!(t)=t +1􀀀 is non-decreasing for < < 1=p; (0.0.9) (Z =(n+1) 0

t􀀀 j (x; t)j sin (t=2) ~!(t) p dt )1=p = Ox 􀀀 (n + 1) 􀀀1=p ; (0.0.10) Z =(n+1)

t􀀀 j (x; t)j sin (t=2) ~!(t) p dt 1=p = Ox 􀀀 (n + 1) 􀀀1=p ; (0.0.11) iii where is an arbitrary number such that 1=p < < + 1=p and p􀀀1 + q􀀀1 = 1: We discussed the case p = 1 separetly and two more theorems are proved for p = 1 (Theorem 2.6.1 and Theorem 2.6.2). In weighted Lp-norm, we prove following theorems: Theorem 2.10.1 Let f be a 2 -periodic function belonging to W(Lp; (t)) with p 1; 0 and let the entries of the lower triangular matrices A (an;k) and B (bn;k) satisfy the conditions (0.0.1) - (0.0.3) of Theorem 2.3.1 with An;n = Bn;n = 1 for n = 0; 1; 2; ::: . Then the degree of approximation of f by BA means of its Fourier series is given by

tBA n (f; x) 􀀀 f(x)

p; = O

( =(n + 1)) + (n + 1)1􀀀 Xn m=0 bn;m m + 1 ! ; provided that the positive non-decreasing function (t) satis es the condition: (t)=t is non{decreasing function for some 0 < < 1: (0.0.12) Theorem 2.10.2 Let f be a 2 -periodic function belonging to W(Lp; (t)) with p 1; 0 and let the entries of the lower triangular matrices A (an;k) and B (bn;k) satisfy the conditions (0:0:1) 􀀀 (0:0:3) of Theorem 2.3.1 with An;n = Bn;n = 1 for n = 0; 1; 2; ::: . Then the degree of approximation of ~ f, conjugate of f; by BA means of its conjugate Fourier series is given by

~tBA n (f; x) 􀀀 ~ f(x)

p; = O

( =(n + 1)) + (n + 1)1􀀀 Xn m=0 bn;m m + 1 ! where (t) and are the same as in Theorem 2:10:1. In Chapter 3, we determine the degree of trigonometric approximation of 2 - periodic functions and their conjugates, in terms of the moduli of continuity associated with them, by matrix means of corresponding Fourier series. We also discuss some analogous results with remarks and corollaries. Theorem 3.3.1 Let f be a 2 -periodic function belonging to the class Lp(!) ; 0 and let T (an;k) be a lower triangular regular matrix with non-negative and non-decreasing (with respect to 0 k n) entries with An;n􀀀 = O (1=t(n + 1)). Then the degree of approximation of f by matrix means of its Fourier series is given by ktn(f; x) 􀀀 f(x)kp = O

1 n + 1 Z 1=(n+1) !(t) t +2 dt

; iv provided that ! is a function of modulus of continuity type such that Z v 0 !(t) t +1 dt = O

!(v) v

; 0 < v < : (0.0.13) Theorem 3.3.2 Let f be a 2 -periodic function belonging to the class Lp(~!) ; 0 and let T (an;k) be a lower triangular regular matrix with non-negative and non-decreasing (with respect to 0 k n) entries with An;n􀀀 = O (1=t(n + 1)). Then the degree of approximation of ~ f, conjugate of f, by matrix means of conjugate Fourier series is given by

~tn(f; x) 􀀀 ~ f(x)

p = O

1 n + 1 Z 1=(n+1) ~!(t) t +2 dt

; provided that ~! is a function of modulus of continuity type such that Z v 0 ~!(t) t +1 dt = O

~!(v) v

; 0 < v < : (0.0.14) Theorem 3.10.1 Let f be a 2 -periodic function belonging to Lip(!(t); p)-class with p 1 and let T (an;k) be a lower triangular regular matrix with nonnegative and non-decreasing (with respect to 0 k n) entries with An;n􀀀 = O (1=t(n + 1)). Then the degree of approximation of f by matrix means of its Fourier series is given by ktn(f; x) 􀀀 f(x)kp = O

1 n + 1 Z 1 =(n+1) !(t) t2+1=p dt

; provided !(t) is a positive non-decreasing function satisfying the following condition: Z v 0 ! (t) t1+1=p dt = O

! (v) v1=p

; 0 < v < : (0.0.15) Theorem 3.10.2 Let f be a 2 -periodic function belonging to Lip(!(t); p)-class with p 1 and let T (an;k) be a lower triangular regular matrix with nonnegative and non-decreasing (with respect to 0 k n) entries with An;n􀀀 = O (1=t(n + 1)). Then the degree of approximation of ~ f, conjugate of f; by matrix means of its conjugate Fourier series is given by

~tn(f; x) 􀀀 ~ f(x)

p = O

1 n + 1 Z 1 =(n+1) !(t) t2+1=p dt

; provided !(t) is a positive non-decreasing function satisfying the condition (0:0:15) of Theorem 3:10:1: v In Chapter 4, we generalize the de nition of Lip ( ; p;w) de ned by Guven [36] to the weighted Lipschitz class Lip( ( ); p;w), where ( ) is a positive non-decreasing function, and determine the degree of approximation of f 2 Lip( ( ); p;w) through matrix means of its trigonometric Fourier series. We note that some earlier results are particular cases of our following result: Theorem 4.2.1 Let 1 < p < 1; w 2 Ap; f 2 Lip( ( ); p;w) and A = (an;k) be a lower triangular regular matrix satisfying one of the following conditions: (i) fan;kg 2 AMDS in k and (n + 1)an;0 = O(1); (ii) fan;kg 2 AMIS in k; (iii) Xn k=0

k

An;0 􀀀 An;k+1 k = O (1=n) : Then kf(x) 􀀀 tn(f; x)kp;w = O( (1=n)); where ( ) is a positive non-decreasing function satisfying (1= ) is an non-decreasing function for some > 0: (0.0.16) Chapter 5 deals with the approximation by triangular matrix means of Walsh{ Fourier series in Lp[0; 1)-space, where fan;kg is almost monotone sequence. We generalize some earlier results [91; 93; 105] under less conditions on an;k: We prove the following: Theorem 5.2.1 Let f 2 Lp[0; 1); 1 p 1: Let T (an;k) be a lower triangular regular matrix with non-negative entries, where n = 2m + k for 1 k 2m and m 1: Then (i) if fan;kg 2 AMIS in k and nan;n = O(1); then ktn(f; x) 􀀀 f(x)kp = O

mX􀀀1 j=0 2jan;2j+1􀀀1!_ p(f; 2􀀀j) + !_ p(f; 2􀀀m) ! ; (ii) if fan;kg 2 AMDS in k; then ktn(f; x) 􀀀 f(x)kp = O

mX􀀀1 j=0 2jan;2j!_ p(f; 2􀀀j) + !_ p(f; 2􀀀m) ! : Theorem 5.2.2 Let f 2 Lip( ; p); > 0 and 1 p 1: Let T (an;k) be a lower triangular regular matrix with non-negative entries, where n = 2m + k for vi 1 k 2m and m 1: Then (i) if fan;kg 2 AMIS in k and nan;n = O(1); then ktn(f; x) 􀀀 f(x)kp = 8>>< >>: O(n􀀀 ) ; if 0 < < 1 O(n􀀀1 log n) ; if = 1 O(n􀀀1) ; if > 1; (ii) if fan;kg 2 AMDS in k; then ktn(f; x) 􀀀 f(x)kp = O

mX􀀀1 j=0 2(1􀀀 )jan;2j + 2􀀀m ! : Chapter 6 deals with the approximation properties of f 2 L[0;1) by Ces aro means of order 1 of the Fourier-Laguerre series of f for any x > 0: We prove the result for x = 0 separately. Theorem 6.2.1 Let f be a function belonging to L[0;1): Then the degree of approximation of f by the Ces aro means of order 1 of the Fourier-Laguerre series of f is given by jC n(f; x) 􀀀 f(x)j = o ( (n)) ; where (t) is a positive non-decreasing function such that (t) ! 1 as t ! 1 and satis es the following conditions: (t) = Z t y =2􀀀1=4j (x; y)jdy = o ( (1=t)) ; t ! 0; (0.0.17) Z t j (x; u)j u du = o ( (1=t)) ; t ! 0; (0.0.18) and Z 1 n e􀀀y=2 y =2􀀀13=12j (x; y)jdy = o 􀀀 n􀀀1=2 (n)

; n ! 1; (0.0.19) where is a xed positive constant and 􀀀1 2 : This holds uniformly for every xed positive interval 0 < x ! < 1: For x = 0; we prove the following theorems: Theorem 6.4.1 Let f be a function belonging to L[0;1): Then the degree of approximation of f at x = 0 by the Ces aro means of order 1 of the Fourier- Laguerre series of f is given by jC n(f; 0) 􀀀 f(0)j = o 􀀀 n =2+3=4 (n)

; vii where (t) is a positive non-decreasing function such that (t) ! 1 as t ! 1 and satis es the conditions (0:0:17) and (0:0:19) of Theorem 6:2:1 for x = 0; > 0 and 2 [􀀀1=2; 1=2]: Theorem 6.7.1 The degree of approximation of f 2 L[0;1) at x = 0 by the Hausdor means of the Fourier-Laguerre series generated by H 2 H1 is given by jHn(f; 0) 􀀀 f(0)j = o( (n)); where (t) is a positive non-decreasing function such that (t) ! 1 as t ! 1 and satis es the following conditions (y) = Z t 0 j'(y)jdy = o 􀀀 t +1 (1=t)

; t ! 0; (0.0.20) Z n

ey=2 y􀀀((2 +3)=4)j'(y)jdy = o 􀀀 n􀀀((2 +1)=4) (n)

; (0.0.21) and Z 1 n ey=2 y􀀀1=3j'(y)jdy = o( (n)); n ! 1; (0.0.22) where is a xed positive constant and > 􀀀1=2: We also discuss some particular cases of Theorem