International Journal of Analysis

Volume 2015, Article ID 478345, 4 pages

http://dx.doi.org/10.1155/2015/478345

## Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

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

Received 6 July 2015; Revised 18 August 2015; Accepted 23 August 2015

Academic Editor: Dong Ye

Copyright © 2015 M. L. Mittal and Mradul Veer Singh. 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 have studied the degree of approximation of a function using different summability (Cesáro means of order : , Euler , and Nörlund ) means of its Fourier-Laguerre series at the point after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by -transform of its Fourier-Laguerre series at frontier point , where is a general lower triangular regular matrix. A particular case, when is a Cesáro matrix of order 1, that is, , has also been discussed as a corollary of main result.

*Dedicated to Professor Bani Singh*

#### 1. Introduction

Let be given infinite series with the sequence of its th partial sums . Define , . If , then the series is said to be -summable to .

Let be an infinite triangular matrix with real constants. The sequence-to-sequence transformation , defines the -transform of the sequence . The series is said to be -summable to if . Throughout this paper, has nonnegative entries with row sums one. is said to be regular if it is limit preserving over the space of convergent sequences. Thus, behaves as a linear operator.

The -transform of , denoted by , are defined by If as , then the series is said to be -summable to . The regularity of method follows from the regularity of method as well as -method and thus the matrix behaves as a linear operator. Some important particular cases of the matrix-Euler operator are as follows:(i)If , then we get operator.(ii)Let be a sequence of real, nonnegative numbers such that , , and . If , then we get operator. A special case in which , ; then operator further reduces to operator.(iii)If , where , then we get operator.(iv)If in above cases, then we get , , , and operators, respectively.(v)If we take identity matrix instead of Euler matrix , then operators reduce to -operators which further reduce to Cesáro , Euler , Harmonic , and Nörlund operators with suitable choice of as above.

*Remark 1. *The product summability methods are more powerful than the individual summability methods; for example, the infinite series is neither -summable nor -summable. However, it can be shown easily that the above series is -summable [1, page 11]. Thus, the product summability methods give an approximation for wider class of functions than the individual methods. Some more examples and recent results on product summability methods can be seen in [2, 3].

*Remark 2. *As in [4], for operator , the product summability operator behaves as double digital filter and thus plays an important role in signal theory.

The Fourier-Laguerre expansion of a function is given by where and denotes the th degree Laguerre polynomial of order , defined by the generating function provided the integral in (3) exists. The elementary properties of Laguerre polynomials can be seen in [5, 6]. Let denote the partial sums, called Fourier-Laguerre polynomials of degree , of the first terms of the Fourier-Laguerre series of in (2). At the point , since and . Thus, using and (1), we get We write

#### 2. Main Results

Various investigators such as Gupta [7], Singh [8], Beohar and Jadiya [9], Lal and Nigam [10], and Nigam and Sharma [11] have studied the degree of approximation of a function using different summability , and methods of series (2) at the point after replacing the continuity condition in Szegö theorem [12] by much lighter conditions. The main aim of this paper is to generalize these earlier results in view of Remark 1. We prove the following.

Theorem 3. *Let be an infinite lower triangular regular matrix with nonnegative entries. Then, the degree of approximation of a function by its Fourier-Laguerre expansion (2) at the point using matrix-Euler operators is given by **provided that **where is a fixed positive constant, , and is a positive monotonic increasing function of such that as .*

Corollary 4 (see [13]). *The degree of approximation of a function by its Fourier-Laguerre expansion (2) at the point using -means is given: **provided (9), (10), and (11) and supplementary conditions on , , and hold as in Theorem 3.*

*Proof. *If , then operator reduces to operator. Hence, the proof is completed.

#### 3. Lemmas

Lemma 5 (see [6, page 177]). *Let be arbitrary and real and and fixed positive constants and let . Then, *

*Lemma 6 (see [6, page 241]). Let and be arbitrary and real, , and . Then, for , where *

*4. Proof of the Main Results*

*4. Proof of the Main Results*

*In view of the orthogonality of Laguerre polynomials [6, page 100] and (6) and (7), where in view of Lemma 5 (first part) and condition (9), andin view of Lemma 5 (second part) and condition (9), integrating by parts and using the argument as in [11, page 6]. Alternatively, using for , (18) can be proved as in [13, pages 37-38]. Now using Lemma 6 and condition (10), and using Lemma 6 and condition (11). Combining (17)–(21) and putting them into (16), this completes the proof of Theorem 3.*

*Conflict of Interests*

*Conflict of Interests*

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

*Acknowledgment*

*Acknowledgment*

*The second author is thankful to the Ministry of Human Resource Development, India, for financial support to carry out this research work.*

*References*

*References*

- V. N. Mishra, V. Sonavane, and L. N. Mishra, “On trigonometric approximation of
*W*(*L*^{p},*ξ*(*t*)) ($p\ge 1$) function by product (*C*,1)(*E*,1) means of its Fourier series,”*Journal of Inequalities and Applications*, vol. 2013, article 300, 12 pages, 2013. View at Publisher · View at Google Scholar - L. N. Mishra, V. N. Mishra, K. Khatri, and Deepmala, “On the trigonometric approximation of signals belonging to generalized weighted Lipschitz $W({L}^{r},\xi (t))(r\ge 1)$-class by matrix $({C}^{1}.{N}_{p})$ operator of conjugate series of its Fourier series,”
*Applied Mathematics and Computation*, vol. 237, pp. 252–263, 2014. View at Publisher · View at Google Scholar - V. N. Mishra, K. Khatri, and L. N. Mishra, “Trigonometric approximation of periodic signals belonging to generalized weighted Lipschitz ${W}^{\prime}({L}_{r},\xi (t))$; $(r\ge 1)$-class by Nörlund-Euler $\left(N,pn\right)(E,q)$ operator of conjugate series of its Fourier series,”
*Journal of Classical Analysis*, vol. 5, no. 2, pp. 91–105, 2014. View at Google Scholar - M. L. Mittal and U. Singh, “$T.{C}_{1}$ summability of a sequence of Fourier coefficients,”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 702–706, 2008. View at Publisher · View at Google Scholar - E. D. Rainville,
*Special Functions*, Macmillan, New York, NY, USA, 1960. View at MathSciNet - G. Szegö,
*Orthogonal Polynomials*, Colloquium Publication, American Mathematical Society, New York, NY, USA, 1975. - D. P. Gupta, “Degree of approximation by Cesáro mean of Fourier-Laguerre expansions,”
*Acta Scientiarum Mathematicarum (Szeged)*, vol. 32, pp. 255–259, 1971. View at Google Scholar - T. Singh, “Degree of approximation by harmonic means of Fourier-Laguerre expansions,”
*Publicationes Mathematicae Debrecen*, vol. 24, no. 1-2, pp. 53–57, 1977. View at Google Scholar · View at MathSciNet - B. K. Beohar and B. L. Jadiya, “Degree of approximation by Cesáro mean of Fourier-Laguerre series,”
*Indian Journal of Pure and Applied Mathematics*, vol. 11, no. 9, pp. 1162–1165, 1980. View at Google Scholar · View at MathSciNet - S. Lal and H. K. Nigam, “Degree of approximation by (N; p; q) summability means of the Fourier-Laguerre expansion,”
*Tamkang Journal of Mathematics*, vol. 32, pp. 143–149, 2001. View at Google Scholar - H. K. Nigam and A. Sharma, “A study on degree of approximation by $(E,1)$ summability means of the Fourier-Laguerre expansion,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 351016, 7 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - G. Szegö, “Beiträge zur Theorie der Laguerreschen Polynome. I: Entwicklungssätze,”
*Mathematische Zeitschrift*, vol. 25, no. 1, pp. 87–115, 1926. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Z. Krasniqi, “On the degree of approximation of a function by (
*C, 1*)(*E, q*) means of its Fourier-Laguerre series,”*International Journal of Analysis and Applications*, vol. 1, no. 1, pp. 33–39, 2013. View at Google Scholar

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