Approximation of functions belonging to L[0, ∞) by product summability means of its Fourier-Laguerre series

In this paper, we have proved the degree of approximation of functions belonging to by Harmonic-Euler means of its Fourier-Laguerre series at . The aim of this paper is to concentrate on the approximation properties of the functions in by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f.


Introduction
Various researchers such as Gupta (1971), Singh (1977), Beohar and Jadia (1980), Lal and Nigam (2001), Nigam and Sharma (2010), Krasniqi (2013) and Sonker (2014) obtained the degree of approximation of L[0, ∞) of the Fourier-Laguerre series by Cesàro, Harmonic, Nörlund, Euler, (C, 1) (E, q), (C, 2)(E, q) and Cesàro means, respectively. The degree of approximation of functions belonging to various classes through trigonometric Fourier approximation using different summability ABOUT THE AUTHORS Kejal Khatri received the PhD in Mathematics from SVNIT, Surat. She is a NBHM post-doctoral fellow under Vishnu Narayan Mishra at SVNIT, Surat. Her research interest is Approximation theory. She has published many research articles in reputed international journals. She is a referee of several international journals in frame of Mathematics. Citations of her research contributions can be found in many scientific journal articles.
Vishnu Narayan Mishra received the PhD in Mathematics from IIT, Roorkee. His research interests are in the areas of pure and applied mathematics. He has published more than 110 research articles in reputed international journals of mathematical and engineering sciences. He is a referee and an editor of several international journals in frame of Mathematics. He guided many postgraduate and PhD students. Citations of his research contributions can be found in many books and monographs, PhD thesis and scientific journal articles.

PUBLIC INTEREST STATEMENT
In this paper, we have used Harmonic-Euler means and determined the degree of approximation of functions with the help of Fourier-Laguerre series at x = 0. Yet, no one has used Harmonic-Euler product summability methods for obtaining the degree of approximation of functions f ∈ L[0, ∞). The paper is interesting and useful from application point of view. Approximates value of many known functions can be evaluated with the help of Fourier-Laguerre series. Research scholars will get motivation through this paper. methods with monotone rows has been proved by many investigators like Khan (1974Khan ( , 1973Khan ( -1974Khan ( , 1982, Mishra (2007), Mishra, Khatri, and Mishra (2012a, 2012b, 2013, , Mishra, Khatri, Mishra, and Deepmala (2014). A number of researchers Liu and Srivastava (2006), Alzer, Karayannakis, and Srivastava (2006), Bor, Srivastava, and Sulaiman (2012, Choi and Srivastava (1991) have proved interesting results in sequences and series using different type of linear summability operators. In Alghamdi and Mursaleen (2013) discussed Hankel matrix transformation of the Walsh-Fourier series and Alotaibi, and Mursaleen (2013) studied on applications of Hankel and regular matrices in Fourier series. In 2014, Mursaleen, and Mohiuddine (2014) discussed convergence methods for double sequences. In this paper, We have extended the previous known results which have already discussed above. 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.
Analysis of signals or time functions are of great importance, because it convey information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Especially, Psarakis, and Moustakides (1997) presented a new L 2 based method for designing the Finite Impulse Response (FIR) digital filters and get corresponding optimum approximations having improved performance.
Let ∑ ∞ n=0 a n be a given infinite series with the sequence of n th partial sums {s n }. Let {p n } be a nonnegative sequence of constants, real or complex, and let us write The series ∑ ∞ n=0 a n is said to be Harmonic (H 1 ) -summable to s, if This method was introduced by Riesz (1924).
The (E, 1) means is defined as the n th partial sum of (E, 1) summability and we denote it by E 1 n . If the series ∑ ∞ n=0 a n is said to be (E, 1) -summable to sum s Hardy (1949).
The product of H 1 summability with a E 1 summability defines H 1 ⋅ E 1 summability. Thus the H 1 ⋅ E 1 mean is given by If t HE n → s as n → ∞, then the infinite series ∑ ∞ n=0 a n is said to be H 1 ⋅ E 1 summable to the sum s.
The Fourier-Laguerre expansion of a function f (x) ∈ L[0, ∞) is given by where and L ( ) n (x) stands the n th degree Laguerre polynomial of order > −1, defined by the generating function provided the integral in (3) exists. The elementary properties of Laguerre polynomials can be seen in Rainville (1960) and Szegö (1975). Let s n (f ; where is a fixed positive constant, ∈ (−1, −1∕2) and (q) is a positive monotonic increasing function of q such that (P n ) → ∞ as P n → ∞ (as n → ∞).

Lemmas
We use the following lemmas in the proof of Theorem 2.1.

Lemma 3.1 Let be an arbitrary real number, a and be fixed positive constants. Then
as P n → ∞ as n → ∞.
Proof The proof is similar as in Szegö (1975, p. 177).
Proof The proof is similar as in Szegö (1975, p. 177).
using Lemma 3.1 (second part) and condition (8), integrating by parts and using the argument as in Krasniqi (2013) and Nigam and Sharma (2010).