SOME PROBLEMS ON TRIGONOMETRIC APPROXIMATION IN Lp ( p ≥1) SPACES

Abstract

The study of approximation properties of the periodic functions in Lp(p ≥ 1)-spaces, in general and in Lipschitz classes Lipα, Lip(α, p), Lip(α, p,w), Lip(ξ(t), p), Lip(ω(t), p), W(Lp,ω(t), β) and W(Lp,Ψ(t), β), p ≥ 1, in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f (x) − Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f . In this thesis, we study the approximation properties of functions belonging to various function classes through trigonometric Fourier series and conjugate trigonometric series. The present thesis is divided into six chapters and the chapterwise description is given below: Chapter 1 is introductory in nature and gives the details of developments in research on the trigonometric Fourier approximation and some basic concepts and definitions. Current status of the field, objective of the work done and layout of the thesis are also given in this chapter. Chapter 2 deals with the approximation properties of the periodic functions and their conjugates belonging to the Lipschitz classes Lipα andW(Lp,ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix (C1.T) means of the Fourier series and conjugate Fourier series, respectively. We prove the following theorems in Chapter 2: Theorem 2.2.1. Let T ≡ (an,k) be a lower triangular regular matrix with non-negative and non-decreasing (with respect to k, for 0 ≤ k ≤ n) entries which satisfies, An,0 = 1, ∀n ∈ N0 and an,n−k − an+1,n+1−k ≥ 0 f or 0 ≤ k ≤ n. (0.1) iii Then the degree of approximation of a 2π-periodic function f ∈ Lipα by C1.T means of its Fourier series is given by ∥tC1.T n ( f ) − f (x)∥∞ = { O((n + 1)−α), 0 < α < 1, O(log(n + 1)/(n + 1)) , α = 1. (0.2) Theorem 2.2.2. Let T ≡ (an,k) be a lower triangular regular matrix same as in Theorem 2.2.1. Then the degree of approximation of a 2π-periodic function f ∈ W(Lp,ω(t), β) with p > 1 and 0 < β < 1/p by C1.T means of its Fourier series is given by ∥ tC1.T n ( f ; x) − f (x) ∥p= O ( (n + 1)βω (1/(n + 1)) ) , (0.3) provided a positive increasing function ω(t) satisfies the following conditions: ω(t)/t is a decreasing function, (0.4) (∫ π/(n+1) 0

ϕ(t) sinβ(t/2)/ω(t)

q dt )1/q = O((n + 1) −1/q), (0.5) (∫ π π/(n+1) ( t−δ|ϕ(t)| sinβ(t/2)/ω(t) )p dt )1/p = O((n + 1)δ−1/p), (0.6) where δ is a real number such that p−1 < δ < β + p−1and p−1 + q−1 = 1. Also conditions (0.5) and (0.6) hold uniformly in x. In the case p = 1, i.e., q = ∞; sup norm is required while using Hölder’s inequality. Therefore, the above proof will not work for p = 1. Thus, for p = 1, we have the following theorem. Theorem 2.5.1. Let T ≡ (an,k) be a lower triangular regular matrix same as in Theorem 2.2.1. Then the degree of approximation of a 2π-periodic function f belonging to the weighted Lipschitz classW(L1,ω(t), β), with 0 < β < 1 by C1.T means of its Fourier series is given by ∥ tC1.T n ( f ; x) − f (x) ∥1= O ( (n + 1)βω(1/(n + 1)) ) , (0.7) provided a positive increasing function ω(t) satisfies (0.4) and the following condition: ω(t)/tβ is non-decreasing, (0.8) ∫ π/(n+1) 0 | ϕ(t) | sinβ(t/2) ω(t) dt = O((n + 1) −1), (0.9) ∫ π π/(n+1) t−δ | ϕ(t) | . sinβ(t/2) ω(t) dt = O((n + 1)δ−1), (0.10) iv where 1 < δ < β + 1. The conditions (0.9) and (0.10) hold uniformly in x. If we replace matrix T with Nörlund matrix, then C1.T means of Fourier series of f reduces to C1.Np means. In the next section, we obtain, in Theorems 2.6.1, 2.6.2 and 2.9.1, the degree of approximation for the function ˜ f , conjugate to the function f belonging to the same classes by C1.T means of the conjugate Fourier series of f . Some corollaries and particular cases are also discussed in this chapter. In Chapter 3, we determine the degree of approximation of ef , conjugate of a 2π- periodic function f belonging to the weighted W(Lp,ω(t), β), p ≥ 1-class by using Hausdorff means of conjugate Fourier series of f . More precisely, we prove: Theorem 3.2.1. Let f be a 2π-periodic function belonging to the weighted Lipschitz class W(Lp,ω(t), β)-class, with p > 1 and 0 ≤ β ≤ 1 − 1/p. Then the degree of approximation of ˜ f by Hausdorff means of conjugate Fourier series of f generated by H ∈ H1,is given by ∥ ˜Hn( f ; x) − ˜ f (x) ∥p= O ( (n + 1)β+1/pξ(1/(n + 1)) ) , (0.11) provided a positive increasing function ξ(t) satisfies the following conditions: ξ(t)/t is non-increasing, (0.12) {∫ π/(n+1) 0 ( | ψx(t) | sinβ(t/2) ξ(t) )p dt }1/p = O((n + 1) −1/p), (0.13) {∫ π/(n+1) ϵ ( ξ(t) t sinβ(t/2) )q dt }1/q = O((n + 1)β+1/pξ(π/(n + 1))), (0.14) {∫ π π/(n+1) ( t−δ | ψx(t) | ξ(t) )p dt }1/p = O((n + 1)δ), (0.15) where δ is an arbitrary number such that 0 < δ < β + 1/p and p−1 + q−1 = 1 for p > 1. The conditions (0.13) and (0.15) hold uniformly in x. Since (C, 1), the Cesàro matrix of order 1, and (E, q), the Euler matrix of order q > 0, are Hausdorff matrices, and the product of two Hausdorff matrices is also a Hausdorff matrix, so the results proved by using product of (C, 1) and (E, q) (q > 0) matrices are particular cases of Theorem 3.2.1. In Theorem 3.4.1, we prove the above result for p = 1. v In Chapter 4, we introduce a more general Lipschitz class Lip(ω(t), p) which includes the classical Lip(ξ(t), p) class of functions i.e., { f ∈ Lp[0, 2π] : ∥ f (x + t) − f (x) ∥p= O(ξ(t)), t > 0 } and { f ∈ Lp[0, 2π] :| f (x + t) − f (x) | = O(t−1/pξ(t)), t > 0 } defined by Khan and Ram and compute analytically the degree of approximation of f ∈ Lip(ω(t), p) using matrix means of the Fourier series of f generated by the matrix T ≡ (an,k). We also discuss an example to show the application of the result. Also, in the corollaries of the theorems of this paper, we observe that the degree of approximation of f ∈ Lip(ξ(t), p) is free from p and sharper than the earlier one. The main results of this chapter are: Theorem 4.2.1. Let T ≡ (an,k) be a lower triangular regular matrix with non-negative and non-decreasing (with respect to k, for 0 ≤ k ≤ n) entries with An,0 = 1. Then the degree of approximation of a 2π-periodic function f ∈ Lip(ω(t), p), with p ≥ 1 by matrix means of its Fourier series is given by ∥ tn( f ; x) − f (x) ∥p= O ( (n + 1)1/p ω (π/(n + 1)) ) , (0.16) provided a positive increasing function ω(t) satisfies the following conditions: ω(t)/tσ is an increasing function for 0 < σ < 1, (0.17) ( ϕ(t) (t−1/pω(t)) ) is a bounded function of t, (0.18) (∫ π π/(n+1) ( ω(t) t1+ 1/p )p dt )1/p = O ( (n + 1) ω ( π n + 1 )) , (0.19) where p−1 + q−1 = 1. Also condition (0.18) holds uniformly in x. In Theorem 4.2.2, we prove (0.16) for hump matrices with the condition (n + 1) maxk{an,k} = O(1). In the next section of this chapter, we define W(Lp,Ψ(t), β)-class, a weighted version of Lip(ω(t), p)-class, with weight function sinβp(x/2) and determine the error of approximation of f ∈ W(Lp,Ψ(t), β) using the same matrix means. More precisely, we prove: Theorem 4.7.1. Let T ≡ (an,k) be a lower triangular regular matrix with non-negative and non-decreasing (with respect to k, for 0 ≤ k ≤ n) entries. Then the degree of approximation of vi a 2π-periodic function f ∈ W(Lp,Ψ(t), β) with 0 ≤ β < 1/p and p ≥ 1 by matrix means of its Fourier series is given by ∥ tn( f ; x) − f (x) ∥p= O ( (n + 1)β+1/p Ψ (π/(n + 1)) ) , (0.20) provided a positive increasing function Ψ(t) satisfies the following conditions: Ψ(t)/tβ+1/p is an increasing function, (0.21) ( ϕ(t) sinβ(t/2) t−1/p Ψ(t) ) is bounded function of t, hold uniformely in x, (0.22) (∫ π π/(n+1) ( Ψ(t) t1+ 1/p+β )p dt )1/p = O ( (n + 1)β+1 Ψ ( π n + 1 )) , (0.23) where p−1 + q−1 = 1. In Theorem 4.7.2, we prove (0.20) by using hump matrices with the condition (n + 1) maxk{an,k} = O(1). We also derive some corollaries from our results. In chapter 5, the approximation properties of the matrix means of trigonometric Fourier series of f belonging to weighted Lipschitz class Lip(α, p,w) with Muckenhoupt weights generated by T ≡ (an,k) under relaxed conditions has been investigated. Our theorem extends some of the previous results pertaining to the degree of approximation of functions in weighted Lipschitz class Lip(α, p,w) and the ordinary Lipschitz class Lip(α, p). The main theorem of this chapter is: Theorem 5.2.1. Let f ∈ Lip(α, p,w), p > 1, w ∈ Ap and let T ≡ (an,k) be an infinite lower triangular regular matrix and satisfies one of the following conditions: (i) 0 < α < 1, {an,k} ∈ AMIS in k, (ii) 0 < α < 1, {an,k} ∈ AMDS in k and (n + 1)an,0 = O(1), (iii) α = 1 and Σn−1 k=0 (n − k) |Δkan,k| = O(1), (iv) α = 1, Σnk =0 |Δkan,k| = O(an,0) with (n + 1)an,0 = O(1), (v) 0 < α < 1, Σn−1 k=0

Δk ( An,0−An,k+1 k+1 ) = O ( 1 n+1 ) . Then ∥ f (x) − τn( f ; x)∥ p,w = O((n + 1) −α), n = 0, 1, 2... (0.24) In Chapter 6, we generalize the notion of Λ-strong convergence of numerical sequences to T-strong convergence (an intermediate notion between bounded variation vii and ordinary convergence), using a lower triangular matrix T = (an,k) with nondecreasing monotone rows of positive numbers tending to ∞ i.e., an,k ≤ an,k+1∀n and limk→∞ an,k = ∞ ∀n. We say, a sequence U = {uk} of complex numbers converges T-strongly to a complex number u if lim n→∞ 1 an,n nΣ k=0 | an,k(uk − u) − an,k−1(uk−1 − u) |= 0 Here an,−1 = 0 and u−1 = 0. We also establish a relationship between ordinary convergence and T-strong convergence. We denote, class all the T-strongly convergent sequences U = {uk} of complex numbers by c(T). Obviously, c(T) is a linear space. Further, we define a norm on c(T) as ∥U∥ c(T) := sup n≥0 1 an,n nΣ k=0 | an,kuk − an,k−1uk−1 | The main result of this chapter is: Lemma 6.1.1. T-strong convergence of a sequence U = {uk} to a number u implies the following two conditions (i) ordinary convergence o f U = {uk} to u, and (0.25) (ii) lim n→∞ 1 an,n nΣ k=1 an,k−1 | uk − uk−1 |= 0, (0.26) and vice-versa. We write σn := 1 an,n nΣ k=0 (an,k − an,k−1)uk (n = 0, 1, ...). Lemma 6.1.2. Convergence of σn to u in the ordinary sense together with (0.26) of Lemma 6.1.1 implies the T-strong convergence of U to number u. Theorem 6.2.1. The class c(T) together with the norm ∥.∥ c(T) is a Banach space. In Theorem 6.2.2, we show that Banach space c(T) has a Schauder basis. We also apply the notion of T-strong convergence on the trigonometric Fourier series under C-metric and Lp-metric.