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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 351016, 7 pages
Research Article

A Study on Degree of Approximation by Summability Means of the Fourier-Laguerre Expansion

Department of Mathematics, Faculty of Engineering & Technology, Mody Institute of Technology and Science, Deemed University, Laxmangarh - 332311, Sikar, Rajasthan, India

Received 12 April 2010; Accepted 26 July 2010

Academic Editor: V. R. Khalilov

Copyright © 2010 H. K. Nigam and Ajay Sharma. 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.


A very new theorem on the degree of approximation of the generating function by means of its Fourier-Laguerre series at the frontier point is obtained.

1. Introduction

Let be an infinite series with the sequence of its th partial sums .

If then we say that is summable by means (see the study by Hardy [1]), and it is written as , where is the sequence of th partial sums of the series .

The Fourier-Laguerre expansion of a function is given by where and denotes the th Laguerre polynomial of order , defined by generating function and existence of integral (1.3) is presumed.

We write Gupta [2] estimated the order of the function by Cesàro means of series (1.2) at the point after replacing the continuity condition in Szegö’s theorem [3] by a much lighter condition. He established the following theorem.

Theorem 1.1. If then provided that , with being the th Cesàro mean of order .

Denoting the harmonic means by , Singh [4] estimated the order of function by harmonic means of series (1.2) at point by weaker conditions than those of Theorem 1.1. He proved the following theorem.

Theorem 1.2. For provided that is a fixed positive constant,

2. Main Theorem

The objects of present paper are as follows:(1)We prove our theorem for means which is entirely different from and harmonic means.(2)We employ a condition which is weaker than condition (1.9) of Theorem 1.2.(3)In our theorem the range of is increased to , which is more useful for application.

In fact, we establish the following theorem.

Theorem 2.1. If then the degree of approximation of Fourier-Laguerre expansion (1.2) at the point by means is given by provided that is a fixed positive constant and , where is a positive monotonic increasing function of such that as .

3. Lemmas

Lemma 3.1 (see the study by Szegö, 1959, [3, page 175]). Let be arbitrary and real, let and be fixed positive constants, and let Then

4. Proof of the Main Theorem

Since therefore, Now, Using orthogonal property of Laguerre’s polynomial and (1.5), we have Using orthogonal property and condition (3.2) (taking for and for ) of Lemma 3.1, we get Further, using orthogonal property and condition (3.1) (taking for , 1 for , and for ) of Lemma 3.1, we get Now, since Therefore, By (4.7) and (4.9), we have, Thus, Now, we consider Finally, Combining (4.4), (4.5), (4.11), (4.12), and (4.13), we get This completes the proof of the theorem.


  1. G. H. Hardy, Divergent Series, University Press, Oxford, UK, 1st edition, 1949. View at MathSciNet
  2. D. P. Gupta, “Degree of approximation by Cesàro means of Fourier-Laguerre expansions,” Acta Scientiarum Mathematicarum, vol. 32, pp. 255–259, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. Szegö, Orthogonal Polynomials, Colloquium Publications American Mathematical Society, New York, NY, USA, 1959.
  4. 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 Zentralblatt MATH · View at MathSciNet