An Estimate of the Rate of Convergence of the Fourier Series in the Generalized Hölder Metric by Delayed Arithmetic Mean

L. Nayak, G. Das, and B. K. Ray 1 School of Applied Sciences, KIIT University, Bhubaneswar, Odisha 751024, India 2Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar 751004, India 3 Department of Mathematics, Ravenshaw University, Cuttack 751007, India Correspondence should be addressed to L. Nayak; laxmipriyamath@gmail.com Received 6 December 2013; Revised 16 April 2014; Accepted 16 April 2014; Published 7 May 2014 Academic Editor: Baruch Cahlon Copyright © 2014 L. Nayak et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the rate of convergence problem of the Fourier series by Delayed Arithmetic Mean in the generalized Hölder metric (H p ) space which was earlier introduced by Das, Nath, and Ray and obtain a sharper estimate of Jackson’s order.

Let Let ( ) be the th partial sum of (1). Then it is known ( [1], page 50) that where ( ) = sin( + 1/2) /2 sin( /2) is known as Dirichlet's kernel. Let when the norm has been taken with respect to throughout the paper. The quantities ( , ) and ( , ) are, respectively, called the modulus of continuity and integral modulus of continuity of . It is known ( [1], page 45) that ( , ) and ( , ) both tend to zero as → 0. It was Prössdorf [2] who first studied the degree of approximation problems of the Fourier series in (0 < ≤ 1) space in the Hölder metric. Generalizing the Hölder metric, Leindler [3] introduced the space given by 2 International Journal of Analysis where is a modulus of continuity; that is, is a positive nondecreasing continuous function on [0, 2 ] with the following property: Further Leindler [3] has introduced the following metric on space: In the case ( ) = , 0 < ≤ 1 the space reduces to space (the norm ‖ ⋅ ‖ being replaced by ‖ ⋅ ‖ ) which is introduced by Prössdorf [2]. It is known that [2] ⊆ ⊆ C 2 , 0 ≤ < ≤ 1.
be an infinite series and let ( ) denote the sequence of its th partial sums. Then the series ∑ is said to be summable ( , ) ( > −1) to the sum (finite), if (see [1], page 76) where and are defined by the following formulae: where (| | < 1), > −1.
From the definition of and it follows that ([1], page 77) The numbers and are called, respectively, the Cesàro sums and the Cesàro means of order ( > −1) of the series ∑ . Applications of the Cesàro transformation can be found in engineering, for example, modal dynamics in earthquake engineering (see Chen and Hong [11], Chen et al. [12]).

Delayed Arithmetic
be an infinite series with sequence of arithmetic mean { }. The Delayed Arithmetic Mean , of ∑ is given by ( [1], page 80) where is a positive integer.
And it is known [1] that which may be called first type Delayed Arithmetic Mean. But in this present paper we take = 2 for Delayed Arithmetic Mean, which is of the form This may be called second type Delayed Arithmetic Mean.
International Journal of Analysis 3 Let ( ; ) and ,2 ( ; ), respectively, denote the first arithmetic mean and second type Delayed Arithmetic Mean of (1). It is known that (see [1], page 88 and page 89) the Fejer's kernel It can be easily verified that We adopt the following additional notations: (25)

Introduction
Extensive investigations of approximation by polynomials and involve mediocre approximation involving (log ) times worse than the best approximation (see the remarks by [1], page 122). The following theorems, in particular, are being quoted to vindicate that (log ) persists in the order of approximation involving and .
Das et al. [9] developed a new space ( ) and generalized the theorem of Leindler [3] to give the following.
Prössdorf [2] obtained the following result concerning the degree of approximation of the Fourier series using Fejer's mean in the Hölder metric.

Main Results
We prove the following theorems.
We need the following lemma.

Remarks and Corollaries
Comparing Theorem 1 with Theorem A and Theorem C we observe that trigonometric polynomial ,2 ( ; ) provides sharper estimation than those obtained by sequence of partial sums of a Fourier series. The comparison of Theorem 2 with Theorem B also exhibits the advantage of ,2 ( ; ) over the sequence of partial sums of the Fourier series in the degree of approximation problem.
Taking = ∞ in Corollary 4, we obtain the following.
For the space Lip(1, 1) Quade has provided the result (32) of Theorem E using ( ; ) trigonometric polynomial. Estimate given in (32) certainly is not of Jackson's order. Therefore in Corollary 6 a more general trigonometric polynomial (namely, ,2 ( ; )) has been taken to ensure Jackson's order, which is sharper than the one given in (32).