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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 495075, 9 pages
http://dx.doi.org/10.1155/2008/495075
Research Article

Error Bound of Periodic Signals in the Hölder Metric

1121 Mahashweta Nagar, Ujjain 456010, India
2Department of Mathematics, Jawaharlal Institute of Technology, Borawan 451228, Dist. Khargone, India

Received 16 October 2007; Revised 16 January 2008; Accepted 9 March 2008

Academic Editor: Narendra Kumar Govil

Copyright © 2008 Tikam Singh and Pravin Mahajan. 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 obtain two theorems to determine the error bound between input periodic signals and processed output signals, whenever signals belong to -space and as a processor we have taken -mean and generalized an early result of Lal and Yadav in (2001).

1. Introduction

Chandra [1] was first to extend Prössdorf’s [2] result to find the degree of approximation of a continuous function using the Nörlund transform. Later on, Mohapatra and Chandra [3] obtained a number of interesting results on the degree of approximation in the Hölder metric using matrix transforms, which generalize all the previous results based on Cesàro and Nörlund transforms. In 1992, Singh [4] introduced -space in place of -space and obtained several results on the degree of approximation of functions and deduced many previous results based on -spaces. In 1996, Das et al. [5] used -space in place of -space and obtained degree of approximation of functions and generalized the results of Mohapatra and Chandra [3]. In 2000, Mittal and Rhoades [6] also obtained the degree of approximation of functions in a normed space and generalized the results of Singh [4] by removing the hypothesis of monotonicity of the rows of the matrix. Singh and Soni [7], and Mittal et al. [8] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals.

2. Definitions and Notations

Let the transforms

A:B: be two regular methods of summability. Then, the A transform of the B transform of a sequence is given by the sequence is said to be summable to the sum , if Let be a -periodic analog signal whose Fourier trigonometric expansion be given by and let be the sequence of partial sums of (2.5).

Let the and transforms for the sequence be defined by respectively.

The product -transform is expressed as the -transform of -transform of and is given by sequence-to-sequence transformation (see, e.g., [9]): The sequence is said to be summable to the sum , if

2.1. Regularity Condition of -Method

whereNow,

(i)(ii) as , for fixed ,(iii) thus, -method is regular.

Singh [4] defined the space by

and the norm by where and choosing , and being increasing signals of . If and , , and being positive constants, then the space is Banach space [2] and the metric induced by the norm on is said to be Hölder metric.

We write

3. Known Result

Lal and Yadav [10] established the following theorem to estimate the error between the input signal and the signal obtained after passing through the -transform.

Theorem A. If a function is -periodic and belonging to class , , then the degree of approximation by means of its Fourier series is given by

4. Main Result

The object of this paper is to generalize the above result under much more general assumptions. We will measure the error between the input signal and the processed output signal , by establishing the following theorems.

Theorem 4.1. Let defined in (2.12) be such that then, for and , we have

Theorem 4.2. Let defined in (2.12) and for and , we have

5. Lemmas

We will use following lemmas.

Lemma 5.1. Let be defined in (2.16), then for , we have It is easy to verify.

Lemma 5.2. Let be defined in (2.17), then where “" is an absolute constant, not necessarily the same at each occurrence.

Proof.

Lemma 5.3.

Proof.

Lemma 5.4 (see [9]). For , then

Lemma 5.5 (see [6]). If satisfies conditions (4.1) and (4.2), then

6. Proof of Theorem 4.1

Proof of Theorem 4.1. Following Zygmund [11], we have From (2.6) and (2.16), we have Using Lemma 5.2, we have Now from (2.8), the -transform of -transform is given by Setting now using (4.1), (4.2), (5.2), and Lemma 5.5, we get Again using (5.2), (4.1), and Lemma 5.3, we have Now from (5.1), Lemmas 5.3 and 5.4, we have Now noting that we have, from (6.6) and (6.8), and from (6.7) and (6.9), we have Thus, from (2.13), (6.11) and (6.12), we have It is to be noted from (6.6) and (6.7), Combining (6.13) and (6.14), we get This completes the proof of Theorem 4.1.

Proof of Theorem 4.2. Follows analogously as the proof of Theorem 4.1 with slight changes, so we omit details.

7. Applications

The following results can easily be derived from the Theorem 4.1. If we put , and replace by and set then we get Corollary 7.1.

Corollary 7.1. If , , then If we put , then from above corollary, we have Corollary 7.2.

Corollary 7.2. If , , then Hence Theorem 3 is particular case of Theorem 4.1.

Acknowledgment

The authors are highly grateful to the referee for his valuable comments and suggestions for the improvement and the better presentation of the paper.

References

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