International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012Β (2012), Article IDΒ 964101, 11 pages

http://dx.doi.org/10.1155/2012/964101

## Trigonometric Approximation of Signals (Functions) Belonging to Class by Matrix Operator

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

Received 22 March 2012; Revised 24 April 2012; Accepted 3 May 2012

Academic Editor: JewgeniΒ Dshalalow

Copyright Β© 2012 Uaday Singh 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

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2*Ο*-periodic signals (functions) belonging to Lip class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip-
class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lipβ*Ξ±* and (, ) classes by using CesΓ‘ro-NΓΆrlund summability without monotonicity condition on , which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).

#### 1. Introduction

For a given signal (function) , , let denote the partial sum, called trigonometric polynomial of degree (or order) , of the first terms of the Fourier series of . Let be a nonnegative sequence of real numbers such that as and .

Define the NΓΆrlund () means of the sequence or Fourier series of . The Fourier series of is said to be NΓΆrlund () summable to if . The Fourier series of is called CesΓ‘ro-NΓΆrlund () summable to if We note that and are also trigonometric polynomials of degree (or order) .

Some interesting applications of the CesΓ‘ro summability can be seen in [1, 2].

The -norm of signal is defined by A signal (function) is approximated by trigonometric polynomials of degree , and the degree of approximation is given by This method of approximation is called trigonometric Fourier approximation.

A signal (function) is said to belong to the class if , and if .

For a positive increasing function and , if , and if .

If , then reduces to , and if , then class coincides with the class . for .

We also write , the greatest integer contained in , , .

#### 2. Known Results

Chandra [3] and Khan [4] have obtained the error estimates in class using monotonicity conditions on the means generating sequence , which was generalized by Leindler [5] to almost monotone weights and by Mittal et al. [6] to general summability matrix. Further, Mittal et al. [7] have extended the results of Leindler [5] to general summability matrix, which in turn generalizes some results of Chandra [3] and Mittal et al. [6]. Recently, Lal [8] has determined the degree of approximation of the functions belonging to and classes using CesΓ‘ro-NΓΆrlund () summability with nonincreasing weights . He proved the following theorem.

Theorem 2.1. *Let be a regular NΓΆrlund method defined by a sequence such that
*

Let be a 2*Ο*-periodic function belonging to , then the degree of approximation of by means of its Fourier series is given by

Theorem 2.2. *If is a 2 Ο-periodic function and Lebesgue integrable on [0, 2Ο] and is belonging to class, then its degree of approximation by means of its Fourier series is given by
*

*provided satisfies the following conditions:*

*where is an arbitrary number such that , , , conditions (2.5) and (2.6) hold uniformly in .*

*Remark 2.3. *In the proof of Theorem 2.1 of Lal [8, page 349], the estimate for is obtained as
Since , the is not needed in (2.2) for the case (cf. [9, page 6870]).

*Remark 2.4. * The author has used monotonicity condition on sequence in the proof of Theorem 2.1 and Theorem 2.2, but not mentioned it in the statements. Further in condition (2.4), is a function of not a sequence.

The condition (2.5) of Theorem 2.2 leads to the divergent integral as and [8, page 349]. Also in [8, pages 349-350], the author while writing the proof of Theorem 2.2 has used in the interval , which is not valid for .

#### 3. Main Results

The observations of Remarks 2.3 and 2.4 motivated us to determine a proper set of conditions to prove Theorems 2.1 and 2.2 without monotonocity on . More precisely, we prove the following theorem.

Theorem 3.1. *Let be the NΓΆrlund summability matrix generated by the nonnegative sequence , which satisfies
**
Then the degree of approximation of a 2 Ο-periodic signal (function) by means of its Fourier series is given by
*

Theorem 3.2. *Let the condition (3.1) be satisfied. Then the degree of approximation of a 2 Ο-periodic signal (function) with by means of its Fourier series is given by
*

*provided positive increasing function satisfies the condition (2.4) and*

*where is an arbitrary number such that , , , conditions (3.4) and (3.5) hold uniformly in .*

*Remark 3.3. *For nonincreasing sequence , we have
Thus condition (3.1) holds for nonincreasing sequence ; hence our Theorems 3.1 and 3.2 generalize Theorems 2.1 and 2.2, respectively.

*Note 1. *Using condition (2.4), we get.

#### 4. Lemmas

For the proof of our Theorems, we need the following lemmas.

Lemma 4.1 (see [10, 5.11]). *If is nonnegative and nonincreasing sequence, then for and for any *

Lemma 4.2 (seeββ[8, page 348]). *For , .*

Lemma 4.3. *If is nonnegative sequence satisfying (3.1), then for ,
*

*Proof. *We have
Now, using , for , we get
Using , for and changing the order of summation, we have
Again using , for , Lemma 4.1, (in view of being positive and for all ) and , we get
Using Abelβs transformation, we get
in view of , for and .

Combining (4.5)β(4.7), we get
in view of (3.1) and .

Finally collecting (4.3), (4.4) and (4.8), we get Lemma 4.3.

*Proof of Theorem 3.1. *We have
Denoting means of by , we write
Now, using Lemma 4.2 and the fact that [10], we have
Using Lemma 4.3, we get
where
Collecting (4.10)β(4.13) and writing , for large values of *n*, we get
Hence,
This completes the proof of Theorem 3.1.

*Proof of Theorem 3.2. *Following the proof of Theorem 3.1, we have
Using HΓΆlderβs inequality, , condition (3.4), Lemma 4.2, and , for , we have
in view of the mean value theorem for integrals, and Note 1.

Similarly, using HΓΆlderβs inequality, Lemma 4.3, , , condition (3.5), and the mean value theorem for integrals, we have
where
in view of increasing nature of , , where lie in , and Note 1.

Collecting (4.16)β(4.19), we get
Hence,
This completes the proof of Theorem 3.2.

#### 5. Corollaries

The following corollaries can be derived from Theorem 3.2.

Corollary 5.1. *If , then for , .*

Corollary 5.2. *If , then for ,
*

Corollary 5.3. *If in Corollary 5.2, then for , (5.1) gives
*

#### 6. Conclusion

Various results pertaining to the degree of approximation of periodic functions (signals) belonging to the Lipchitz classes have been reviewed and the condition of monotonocity on the means generating sequence has been relaxed. Further, a proper set of conditions have been discussed to rectify the errors pointed out in Remarks 2.3 and 2.4.

#### Acknowledgment

The authors are very grateful to the reviewer for his kind suggestions for the improvement of this paper.

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