Approximation of functions in the generalized Zygmund class using Hausdorff means

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z_{p}^{(\omega)}$\end{document}Zp(ω) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p \ge1$\end{document}p≥1) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results.


Introduction
During the last few decades the degree of approximation of functions belonging to various Lipschitz classes (Lip α, Lip(α, p), Lip(ξ (t), p), W (L p , ξ (t)) etc.) has been studied by various investigators (see [-] and the references therein) using different summability methods such as Cesàro, Hölder, Euler and their products. Each of the matrices involved in these methods is a Hausdorff matrix and the product of two Hausdorff matrices is again a Hausdorff matrix [-]. Also, multiplication of two Hausdorff matrices is commutative. Thus in view of these remarks, Rhoades [-], Singh, Srivastava [] have obtained the degree of approximation of functions belonging to various Lipschitz classes using Hausdorff means. To the best of our knowledge, the approximation of functions belonging to the generalized Zygmund class by Hausdorff means has not been investigated so far. This motivated us to work in this direction. Here we recall some definitions for our purpose. Let For a given function f ∈ L p , let denote the partial sums, called trigonometric polynomials of degree (or order) n, of the first (n + ) terms of the Fourier series of f at a point x.
Note  Here we shall use the notations of Lal et al. [].
Let ω : [,π] → R be an arbitrary function with ω(t) >  for  < t ≤ π and lim t→ + ω(t) = ω() = . As in [], define  []. Throughout this paper ω and v will denote the Zygmund moduli of continuity such that ω(t)/v(t) is positive and nondecreasing in t. For well-known properties of modulus of continuity and modulus of smoothness, in our case Zygmund modulus of continuity, one can refer the excellent monograph of Zygmund ([], pp.-).
Note that Note  (i) If ω(t) = t α , then the Z (ω) and Z (ω) p classes reduce to the Z α and Z α,p classes, respectively.
(ii) If we take p → ∞, then the Z (ω) p class reduces to the Z (ω) class and the Z α,p class reduces to the Z α class.

Preliminaries
Hausdorff matrices were first introduced by Hurwitz and Silverman () as the collection of lower triangular matrices that commute with the Cesàro matrix of order one. Hausdorff () reintroduced this class in the process of solving the moment problem over a finite interval []. A Hausdorff matrix H ≡ (h n,k ) is an infinite lower triangular matrix [] defined by where the operator is defined by μ n ≡ μ nμ n+ and k+ μ n ≡ k ( μ n ).
A Hausdorff matrix is regular iff where the mass function γ (u) is continuous at u =  and belongs to BV [, ] such that γ () = , γ () = ; and for  < u < , γ (u) = [γ (u + ) + γ (u -)]/. In this case {μ n } has the representation μ n =   u n dγ (u). A detailed discussion and further interesting properties of Hausdorff matrices can be seen in [-, , ] and the references given there.
The Hausdorff means of a trigonometric Fourier series of f is defined by The Fourier series of a function f is said to be summable to s by Hausdorff means if H n (x) → s as n → ∞.
We need the following lemmas in the proofs of our theorems.
Proof We can write since | sin kt| ≤ k| sin t| ∀k ∈ N.
using | sin t| ≤  for all t and (sin t) - ≤ π/t for  < t ≤ π/. This completes the proof of Lemma .
Proof The proof follows directly using (sin t) - ≤ π/t for  < t ≤ π/ and Lemma .  In this paper we prove two theorems on approximation of a function by Hausdorff means of its Fourier series in terms of Zygmund modulus of continuity.

Main result and discussion
Theorem  Let f be a π periodic function, Lebesgue integrable on [-π, π] and belonging to the generalized Zygmund class Z (ω) p , p ≥ . Then the degree of approximation of a signal (function) f , using Hausdorff means of its Fourier series, is given by where ω and v denote the Zygmund moduli of continuity such that ω(t)/v(t) is positive and non-decreasing.

Theorem  In addition to the conditions of Theorem , if t - ω(t)/v(t) is non-increasing, then the degree of approximation of a signal (function) f in Z (ω) p (p ≥ ), using Hausdorff means of its Fourier series, is given by
Now, in view of Note , we present a corollary of our results.
Corollary  Let f be a π periodic function, Lebesgue integrable on [-π, π] and belonging to the Zygmund class Z α,p , p ≥ . Then the degree of approximation of a signal (function) f , using Hausdorff means of its Fourier series, is given by Proof The result can be obtained by putting ω(t) = t α and v(t) = t β in Theorems  and .

Remark 
where a = /( + q), q > , it is easy to verify that μ k = /( + q) k and Thus the Hausdorff matrix H ≡ (h n,k ) reduces to the Euler matrix (E, q) of order q > .
In view of the above remark, our results also hold for different summability methods such as Cesàro, Hölder, Euler and their products.

Using Lemma , Lemma  {part(iii)} and the monotonicity of ω(t)/v(t) with respect to t,
we have Using Lemma  and Lemma  {part(iii)}, we get Thus, from (.), (.) and (.), l n (· + y) + l n (·y) -l n (·) p Again using Lemmas  and , we have This completes the proof of Theorem .

Proof of Theorem 2
Following the proof of Theorem , we have From the assumption that t - ω(t)/v(t) is positive and non-increasing with t, we have This completes the proof of Theorem .

Conclusions
Analysis and approximation of signals (or functions) are of great importance in science and engineering because a signal conveys the attribute of some physical phenomenon. Functions in L p (p ≥ )-spaces are assumed to be most appropriate for practical purposes; for example, L  , L  and L ∞ are of particular interest for engineers in digitization. Fourier methods are commonly used for signal analysis and system design in modern telecommunications, radar and image processing systems. The theory of classical Fourier analysis can be extended to discrete time signals and leads to many effective algorithms that can be directly implemented on general computers or special purpose digital signal processing devices. Thus the study of error estimate of functions in various function spaces such as Lipschitz, Holder, Zygmund, Besov spaces etc. using some summability means of trigonometric Fourier series, also known as trigonometric Fourier approximation (TFA) in the literature, has received a growing interest of investigators over the past few decades. The scientists and engineers use the properties of TFA in designing digital filters. A few more applications have been mentioned in Section . The problem of determining the order of best approximation plays a very important role in approximation theory. In this paper we compute the best approximation of a function f in the generalized Zygmund class Z ω p (p ≥ ) by Hausdorff means of partial sums of trigonometric Fourier series of f . We also deduce a corollary for the Z α,p class. Further we observe that our estimates are analogous to those of Hölder classes. Since, in view of Remark , the popular summability methods such as Cesàro, Hölder, Euler and their product methods are particular cases of Hausdorff method, so our results also hold for these methods.
Besov spaces serve to generalize more elementary function spaces and are effective in measuring the smoothness properties of functions. The best approximation of functions in Besov spaces by Hausdorff means may be the future interest of a few investigators in the direction of this work.