International Journal of AnalysisVolume 2014 (2014), Article ID 171675, 7 pageshttp://dx.doi.org/10.1155/2014/171675

Research Article

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

L. Nayak,¹ G. Das,² and B. K. Ray³

¹School of Applied Sciences, KIIT University, Bhubaneswar, Odisha 751024, India²Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar 751004, India³Department of Mathematics, Ravenshaw University, Cuttack 751007, India

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.

Abstract.

We study the rate of convergence problem of the Fourier series by Delayed Arithmetic Mean in the generalized Hölder metric space which was earlier introduced by Das, Nath, and Ray and obtain a sharper estimate of Jackson's order.

1. Definition

Let be a -periodic function such that . Let the Fourier series of at be given by Let Let be the th partial sum of (1). Then it is known ([1], page 50) that where is known as Dirichlet’s kernel.

Let denote the Banach space of all -periodic continuous functions defined on under the supremum norm. The space with reduces to defined over . We write 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 .

It was Prössdorf [2] who first studied the degree of approximation problems of the Fourier series in space in the Hölder metric. Generalizing the Hölder metric, Leindler [3] introduced the space given by where is a modulus of continuity; that is, is a positive nondecreasing continuous function on with the following property: (i),(ii),(iii).

Further Leindler [3] has introduced the following metric on space:

In the case , the space reduces to space (the norm being replaced by ) which is introduced by Prössdorf [2]. It is known that [2]

The degree of approximation problem in space has been studied by Leindler [3], Totik [4, 5], Mazhar and Totik [6], and Mazhar [7, 8]. The space was further generalized by Das et al. [9] as follows.

For , , we write where is a modulus of continuity. If then we say that Lip. We define It can be seen that is a norm in . To prove the completeness of the space we use the completeness of .

If we put , then reduces to space (with the norm replaced by ) which is introduced earlier by Das et al. [10]. If as , then exists and is 0 everywhere and is constant. Given the spaces and , if is nondecreasing, then since For , if we put and , then (12) reduces to the following: Note that the space is the familiar space introduced earlier by Prössdorf [2].

1.1. The Cesàro Transformation

Let be an infinite series and let denote the sequence of its th partial sums. Then the series is said to be summable to the sum (finite), if (see [1], page 76) where and are defined by the following formulae: where , .

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 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]).

1.2. Delayed Arithmetic Mean

Let 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 For , which may be called first type Delayed Arithmetic Mean.

But in this present paper we take for Delayed Arithmetic Mean, which is of the form This may be called second type Delayed Arithmetic Mean.

Let and , 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:

2. Introduction

Extensive investigations of approximation by polynomials and involve mediocre approximation involving times worse than the best approximation (see the remarks by [1], page 122). The following theorems, in particular, are being quoted to vindicate that 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.

Theorem A (see [9]). Let and be moduli of continuity such that is nondecreasing. If , , then

Theorem B (see [9]). If Lip , , and is of monotonic type, then where is a modulus of continuity.

Recently Leindler [13] proved the following.

Theorem C ([13, Theorem 1]). Let and be moduli of continuity such that is nondecreasing; moreover, for some , let the function be nonincreasing. If , , then

Leindler [13] has derived Theorem C from Theorem A by making use of the fact that whenever is nonincreasing.

Quade [14] however proved the following.

Theorem D ([14, Theorem 5]). Let Lip ; then (i),(ii).

Theorem E ([14, Theorem 6]). If Lip , then(i)if or if , , (ii)if ,

Prössdorf [2] obtained the following result concerning the degree of approximation of the Fourier series using Fejer’s mean in the Hölder metric.

Theorem F (see [2]). Let and ; then

The objective of the present paper is to establish theorems, which involve Jackson’s order (see [15], page 56) not involving .

3. Main Results

We prove the following theorems.

Theorem 1. Let and be moduli of continuity such that is nondecreasing. If , then If in addition is nonincreasing, then and a fortiori

Theorem 2. If Lip , then If in addition is nonincreasing, then and a fortiori

We need the following lemma.

Lemma 3 ([9, Lemma 1]). Let and be defined as in Theorem 1. Then for and (i), (ii)(iii), (iv), where .

4. Proof of Theorem 1

It may be easily verified that and hence By definition Hence Since now, by the generalized Minkowski inequality and for , we have Now, Since is bounded and by Lemma 3, we get Now, Replacing with in (48), we obtain From (48) and (49), it follows that Since , , it follows that as and .

Since , for , it follows that Now, Since , for , using Lemma 3 (iv), we get Similarly, Collecting the results from (45)–(55), we obtain Proceeding in the similar lines, it can be proved that Hence collecting results of (42), (56), and (57), we obtain which ensures (I) of Theorem 1.

If in addition is nonincreasing, we have where is some positive constant. Further Using (59) and (60) in (58), we get (II) and (III) of Theorem 1. This completes the proof.

5. Proof of Theorem 2

We adopt the technique used in the proof of Theorem 1 and use the fact that whenever Lip,, to prove Theorem 2.

6. Remarks and Corollaries

Comparing Theorem 1 with Theorem A and Theorem C we observe that trigonometric polynomial 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 over the sequence of partial sums of the Fourier series in the degree of approximation problem.

We specialize and to obtain the following corollaries from Theorems 1 and 2. Taking and , in Theorem 1, we get the following.

Corollary 4. If , then

Taking in Corollary 4, we obtain the following.

Corollary 5. If , , then

The above corollary can be compared with Theorem F.

Taking , in Theorem 2, we obtain the following.

Corollary 6. If , and , then

For the space Lip 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, ) has been taken to ensure Jackson’s order, which is sharper than the one given in (32).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for his valuable suggestions and criticism which led to the improvement of the paper.

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