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

Abstract

In this paper, we have proved the degree of approximation of functions belonging to $L[0,\infty )$ by Harmonic-Euler means of its Fourier-Laguerre series at $x=0$. The aim of this paper is to concentrate on the approximation properties of the functions in $L[0,\infty )$ by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f.

Keywords:

• degree of approximation
• Harmonic-Euler means
• Fourier-Laguerre series
• orthogonal polynomials and special functions

Mathematics subject classifications:

• 40C05
• 40D25
• 40G05
• 41A25
• 42A10

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 $f\in L[0,\infty )$. 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.

1. 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 $L[0,\infty )$ of the Fourier-Laguerre series by Ces$\stackrel{`}{a}$ro, Harmonic, Nörlund, Euler, (C, 1)(E, q), (C, 2)(E, q) and Ces$\stackrel{`}{a}$ro means, respectively. The degree of approximation of functions belonging to various classes through trigonometric Fourier approximation using different summability methods with monotone rows has been proved by many investigators like Khan (1974,1973–1974,1982), Mishra (2007), Mishra, Khatri and Mishra (2012a,2012b,2013), Mishra and Khatri (2014), 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 ${\sum }_{n=0}^{\infty }{a}_n$ be a given infinite series with the sequence of $n^{th}$ partial sums $\{s_n\}$. Let $\{p_n\}$ be a non-negative sequence of constants, real or complex, and let us write$$\begin{array}{c}\hfill P_n=\sum _{k=0}^n{p}_k\ne 0\forall n\ge 0,p_{-1}=0=P_{-1}\text{and}{P}_n\to \infty \text{as}n\to \infty .\end{array}$$

The series ${\sum }_{n=0}^{\infty }{a}_n$ is said to be Harmonic $(H_1)$ - summable to s, if$$\begin{array}{c}\hfill H_n^1=\frac{1}{\text{log}n}\sum _{k=0}^n\frac{s_k}{(n-k+1)}\to s,\text{as}n\to \infty .\end{array}$$

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 $E_n^1$. If$$\begin{array}{c}\hfill E_n^1=\frac{1}{2^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)s_k\to s,\text{as}n\to \infty ,\end{array}$$

the series ${\sum }_{n=0}^{\infty }{a}_n$ 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(1) $$\begin{array}{c}\hfill t_n^{HE}=\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)}{E}_n^k=\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)s_{\mathit{\nu }}.\end{array}$$(1)

If $t_n^{HE}\to s$ as $n\to \infty$, then the infinite series ${\sum }_{n=0}^{\infty }{a}_n$ is said to be $H_1·E^1$ summable to the sum s.

The Fourier-Laguerre expansion of a function $f(x)\in L[0,\infty )$ is given by(2) $$\begin{array}{c}\hfill f(x)\sim \sum _{n=0}^{\infty }{b}_n(f)L_n^{(\mathit{\beta })}(x),\end{array}$$(2)

where(3) $$\begin{array}{cc}\hfill L_n^{(\mathit{\beta })}(x)& =\frac{1}{n!}{e}^x{x}^{-\mathit{\beta }}\frac{{\text{d}}^n}{\text{d}{x}^n}[x^{n+\mathit{\beta }}{e}^{-x}],\hfill \\ \hfill b_n(f)& ={\left(\mathrm{\Gamma }(\mathit{\beta }+1)\left(\begin{array}{c}n+\mathit{\beta }\\ \mathit{\beta }\end{array}\right)\right)}^{-1}\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}f(z)L_n^{(\mathit{\beta })}(z)\text{d}z,\hfill \end{array}$$(3)

and $L_n^{(\mathit{\beta })}(x)$ stands the $n^{th}$ degree Laguerre polynomial of order $\mathit{\beta }>-1$, defined by the generating function(4) $$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{L}_n^{(\mathit{\beta })}(x){\mathit{\gamma }}^n={(1-\mathit{\gamma })}^{-\mathit{\beta }-1}{e}^{-\left(\frac{x\mathit{\gamma }}{\mathit{\gamma }-1}\right)},\end{array}$$(4)

provided the integral in (3) exists. The elementary properties of Laguerre polynomials can be seen in Rainville (1960) and Szegö (1975). Let $s_n(f;x)={\sum }_{k=0}^n{b}_k(f)L_k^{(\mathit{\beta })}(x),$ denote the partial sums, called Fourier-Laguerre polynomials of degree n, of the first $(n+1)$ terms of the Fourier-Laguerre series of f in (4). At the point $x=0$,$$\begin{array}{cc}\hfill s_n(f;0)& =\sum _{k=0}^n{b}_k(f)L_k^{(\mathit{\beta })}(0)=\frac{1}{\mathrm{\Gamma }(\mathit{\beta }+1)}\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}f(z)\sum _{k=0}^n{L}_k^{(\mathit{\beta })}(z)\text{d}z\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(\mathit{\beta }+1)}\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}f(z)L_n^{(\mathit{\beta }+1)}(z)\text{d}z,\hfill \end{array}$$

since $L_k^{(\mathit{\beta })}(0)=\left(\begin{array}{c}k+\mathit{\beta }\\ \mathit{\beta }\end{array}\right)$ and ${\sum }_{k=0}^n{L}_k^{\mathit{\beta }}(z)=L_n^{(\mathit{\beta }+1)}(z)$. Thus using $s_n(f;0)$ and (1), we get(5) $$\begin{array}{c}\hfill t_n^{HE}(f;0)=\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)\mathrm{\Gamma }(\mathit{\beta }+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}f(z)L_n^{(\mathit{\beta }+1)}(z)\text{d}z.\end{array}$$(5)

We write(6) $$\begin{array}{c}\hfill \mathit{\psi }(z)=\frac{z^{\mathit{\beta }}{e}^{-z}[f(z)-f(0)]}{\mathrm{\Gamma }(\mathit{\beta }+1)}.\end{array}$$(6)

2. Main result

The degree of approximation of functions belonging to $L[0,\infty )$ by different matrix summability methods using Fourier-Laguerre expansion (2) at the point $x=0$ has been determined by various investigators such as Gupta (1971), Singh (1977), Beohar and Jadia (1980), Lal and Nigam (2001), Nigam and Sharma (2010), Krasniqi (2013) and Sonker (2014). But till now, nothing seems to have been done so far to obtain the degree in approximation of functions $f\in L[0,\infty )$ by its using Fourier-Laguerre expansion (2) at the point $x=0$ using Harmonic-Euler summability methods with a suitable set of conditions and prove the following theorem:

Theorem 2.1

If $\{p_n\}$ is a positive non-increasing sequence of real number and the degree of approximation of Fourier-Laguerre expansion (2) at the point $x=0$ using Harmonic-Euler sumability means is given by(7) $$\begin{array}{c}\hfill t_n^{HE}(f;0)-f(0)=o(\mathit{\eta }(P_n)),\end{array}$$(7)

provided that(8) $$\begin{array}{cc}\hfill \mathrm{\Psi }(q)& =\underset{0}{\overset{q}{\int }}|\mathit{\psi }(z)|\text{d}z=o(q^{\mathit{\beta }+1}\mathit{\eta }(1/q),q\to 0,\hfill \end{array}$$(8) (9) $$\begin{array}{cc}\hfill \underset{\mathit{\alpha }}{\overset{P_n}{\int }}{e}^{z/2}{z}^{-(2\mathit{\beta }+3)/4}|\mathit{\psi }(z)|\text{d}z& =o(P_n^{-(2\mathit{\beta }+1)/4}\mathit{\eta }(P_n)),\hfill \end{array}$$(9) (10) $$\begin{array}{c}\hfill \underset{P_n}{\overset{\infty }{\int }}{e}^{z/2}{z}^{-1/3}|\mathit{\psi }(z)|\text{d}z=o(\mathit{\eta }(P_n)),P_n\to \infty \text{as}n\to \infty ,\end{array}$$(10) (11) $$\begin{array}{cc}& \{\mathit{\eta }(q)/q\}\text{is non-increasing in}q.\hfill \end{array}$$(11)

where $\mathit{\alpha }$ is a fixed positive constant, $\mathit{\beta }\in (-1,-1/2)$ and $\mathit{\eta }(q)$ is a positive monotonic increasing function of q such that $\mathit{\eta }(P_n)\to \infty$ as $P_n\to \infty$ (as $n\to \infty$).

Note 1. Using condition (11), we get the inequality: $\mathit{\eta }\left(\frac{\mathit{\pi }}{P_n}\right)\le \mathit{\pi }\mathit{\eta }\left(\frac{1}{P_n}\right)$, for $\left(\frac{\mathit{\pi }}{P_n}\right)\ge \left(\frac{1}{P_n}\right)$.

3. Lemmas

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

Lemma 3.1

Let $\mathit{\beta }$ be an arbitrary real number, a and $\mathit{\alpha }$ be fixed positive constants. Then(12) $$\begin{array}{c}\hfill L_n^{(\mathit{\beta })}(x)=\left\{\begin{array}{cc}O(P_n^{\mathit{\beta }}),\hfill & \text{if}0\le x\le a/P_n,\hfill \\ O(x^{-(2\mathit{\beta }+1)/4}{P}_n^{(2\mathit{\beta }-1)/4}),\hfill & \text{if}a/P_n\le x\le \mathit{\alpha },\hfill \end{array}\right.\end{array}$$(12)

as $P_n\to \infty$ as $n\to \infty$.

Proof

The proof is similar as in Szegö (1975, p. 177).

Lemma 3.2

Let $\mathit{\beta }$ be an arbitrary real number, $\mathit{\alpha }>0$ and $0<\mathit{\xi }<4.$ Then(13) $$\begin{array}{c}\hfill \text{max}{e}^{-x/2}{x}^{(\mathit{\beta }/2+1/4)}|L_n^{(\mathit{\beta })}(x)|=\left\{\begin{array}{cc}O(P_n^{(\mathit{\beta }/2-1/4)}),\hfill & \text{if}\mathit{\alpha }\le x\le (4-\mathit{\xi })P_n,\hfill \\ O(P_n^{(\mathit{\beta }/2-1/12)},\hfill & \text{if}x\ge \mathit{\alpha },\hfill \end{array}\right.\end{array}$$(13)

as $P_n\to \infty$ as $n\to \infty$.

Proof

The proof is similar as in Szegö (1975, p. 177).

Proof of theorem (2.1) (14) $$\begin{array}{cc}\hfill t_n^{HE}(f;0)-f(0)& =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\left(\frac{1}{\mathrm{\Gamma }(\mathit{\beta }+1)}\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}f(z)L_n^{(\mathit{\beta }+1)}(z)\text{d}z-f(0)\right)\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\left(\frac{1}{\mathrm{\Gamma }(\mathit{\beta }+1)}\underset{0}{\overset{\infty }{\int }}{z}^{\mathit{\beta }}{e}^{-z}(f(z)-f(0))L_n^{(\mathit{\beta }+1)}(z)\text{d}z\right)\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{0}{\overset{\infty }{\int }}\mathit{\psi }(z)L_n^{(\mathit{\beta }+1)}(z)\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\left(\underset{0}{\overset{a/P_n}{\int }}+\underset{a/P_n}{\overset{\mathit{\alpha }}{\int }}+\underset{\mathit{\alpha }}{\overset{P_n}{\int }}+\underset{P_n}{\overset{\infty }{\int }}\right)\mathit{\psi }(z)L_n^{(\mathit{\beta }+1)}(z)\text{d}z\hfill \\ & =\sum _{i=0}^4{I}_i,\text{say,}\hfill \end{array}$$(14)

where(15) $$\begin{array}{cc}\hfill |I_1|& \le \frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{0}{\overset{a/P_n}{\int }}|\mathit{\psi }(z)||L_n^{(\mathit{\beta }+1)}(z)|\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)O(P_k^{(\mathit{\beta }+1)})o\left(\frac{a^{\mathit{\beta }+1}\mathit{\eta }(P_n/a)}{P_n^{\mathit{\beta }+1}}\right)\hfill \\ & =O(P_n^{(\mathit{\beta }+1)})o\left(\frac{a^{\mathit{\beta }+1}\mathit{\eta }(P_n/a)}{P_n^{\mathit{\beta }+1}}\right)\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\hfill \\ & =o(\mathit{\eta }((P_n/a)))=o(\mathit{\eta }(P_n)),\hfill \end{array}$$(15)

using Lemma 3.1 (first part) and condition (8).(16) $$\begin{array}{cc}\hfill |I_2|& \le \frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{a/P_n}{\overset{\mathit{\alpha }}{\int }}|\mathit{\psi }(z)||L_n^{(\mathit{\beta }+1)}(z)|\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)O(P_k^{(2\mathit{\beta }+1)/4})\underset{a/P_n}{\overset{\mathit{\alpha }}{\int }}|\mathit{\psi }(z)|z^{-(2\mathit{\beta }+3)/4}\text{d}z\hfill \\ & =O(P_n^{(2\mathit{\beta }+1)/4})\underset{a/P_n}{\overset{\mathit{\alpha }}{\int }}|\mathit{\psi }(z)|z^{-(2\mathit{\beta }+3)/4}\text{d}z\hfill \\ & =o(\mathit{\eta }(P_n)),\hfill \end{array}$$(16)

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).(17) $$\begin{array}{cc}\hfill |I_3|& \le \frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{\mathit{\alpha }}{\overset{P_n}{\int }}{e}^{z/2}{z}^{-(2\mathit{\beta }+3)/4}|\mathit{\psi }(z)|e^{-z/2}{z}^{(2\mathit{\beta }+3)/4}|L_n^{(\mathit{\beta }+1)}(z)|\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)O(P_k^{(2\mathit{\beta }+1)/4})\underset{\mathit{\alpha }}{\overset{P_n}{\int }}{e}^{z/2}{z}^{-(2\mathit{\beta }+3)/4}|\mathit{\psi }(z)|\text{d}z\hfill \\ & =O(P_n^{(2\mathit{\beta }+1)/4})o(P_n^{-(2\mathit{\beta }+1)/4})o(\mathit{\eta }(P_n))\hfill \\ & =o(\mathit{\eta }(P_n)),\hfill \end{array}$$(17)

using Lemma 3.2 and condition (9).(18) $$\begin{array}{cc}\hfill |I_4|& \le \frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)\underset{P_n}{\overset{\infty }{\int }}{e}^{z/2}{z}^{-(3\mathit{\beta }+5)/6}|\mathit{\psi }(z)|e^{-z/2}{z}^{(3\mathit{\beta }+5)/6}|L_n^{(\mathit{\beta }+1)}(z)|\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)O(P_k^{(\mathit{\beta }+1)/2})\underset{P_n}{\overset{\infty }{\int }}{e}^{z/2}{z}^{-(3\mathit{\beta }+5)/6}|\mathit{\psi }(z)|\text{d}z\hfill \\ & =\frac{1}{\text{log}n}\sum _{k=0}^n\frac{1}{(n-k+1)2^k}\sum _{\mathit{\nu }=0}^k\left(\begin{array}{c}k\\ \mathit{\nu }\end{array}\right)O(P_k^{(\mathit{\beta }+1)/2})\underset{P_n}{\overset{\infty }{\int }}\frac{e^{z/2}{z}^{-1/3}|\mathit{\psi }(z)|}{z^{(\mathit{\beta }+1)/2})}\text{d}z\hfill \\ & =O(P_n^{(\mathit{\beta }+1)/2})o(P_n^{-(\mathit{\beta }+1)/2})o(\mathit{\eta }(P_n))\hfill \\ & ==o(\mathit{\eta }(P_n)),\hfill \end{array}$$(18)

using Lemma 3.2 and condition (10), combining (15)–(18) and putting into (14). The proof of the theorem is completed.

Acknowledgements

The authors would like to express their deep gratitude to the anonymous learned referee(s) and Professor H. M. Srivastava, Senior Editor of the journal for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. The first author Kejal Khatri acknowledges the Department of Atomic Energy, National Board Higher Mathematics, Mumbai, India for supporting this research article, DAE Ref. Number: 2/40(58)/2015/R&D-II/13262. The second author Vishnu Narayan Mishra acknowledges that this project was supported by the Cumulative Professional Development Allowance(CPDA), SVNIT, Surat (Gujarat), India.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Kejal Khatri

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

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.