Pointwise summability of Fourier-Laguerre series of integrable functions

TL;DR

This paper extends classical results on the summability of Fourier-Laguerre series for integrable functions, providing new approximation results and illustrative examples.

Contribution

It offers an approximation version of historical summability results for Fourier-Laguerre series, enhancing understanding of their convergence properties.

Findings

New approximation results for Fourier-Laguerre series

Corollaries and examples illustrating the theory

Extension of classical summability theorems

Abstract

We present an approximation version of the results of D. P. Gupta [ J. of Approx. Theory, 7 (1973), 226-238] A. N. S. Singroura [Proc. Japan Acad., 39 (4) (1963), 208-210] and G. Szeg\"{o} [Math. Z., 25 (1926), 87-115]. Some corollaries and examples will also be given.

Equations

$${\int }_{{}_{{}_{{\mathbb{R}}^{+}}}}{e}^{-t}{t}^{\alpha }\mid f(t)\mid dt<\infty .$$

$$S^{\left(\alpha \right)}f\left(x\right):=\sum _{\nu =0}^{\infty }{a}_{\nu }^{\left(\alpha \right)}(f)L_{\nu }^{\left(\alpha \right)}\left(x\right)\text{, with }\alpha >-1,$$

$$L_n^{\left(\alpha \right)}\left(x\right)=\frac{x^{-\alpha }{e}^x}{n!}\frac{d^n}{d{x}^n}\left(x^{n+\alpha }{e}^{-x}\right)=\sum _{\nu =0}^n\frac{\left(-1^{\nu }\right)}{\nu !}\left(\genfrac{}{}{0ex}{}{n+\alpha }{n-\nu }\right)x^{\nu }$$

$$a_{\nu }^{\left(\alpha \right)}(f)=\frac{1}{\Gamma \left(\alpha +1\right)A_n^{\left(\alpha \right)}}{\int }_0^{\infty }{e}^{-y}{y}^{\alpha }{L}_{\nu }^{\left(\alpha \right)}\left(y\right)f\left(y\right)dy,\text{ with }{A}_n^{\left(\alpha \right)}=\left(\genfrac{}{}{0ex}{}{n+\alpha }{n}\right).$$

$$S_k^{\left(\alpha \right)}f\left(x\right)=\sum _{\nu =0}^n{a}_{\nu }^{\left(\alpha \right)}(f)L_{\nu }^{\left(\alpha \right)}\left(x\right)$$

$$S_n^{\left(\gamma ,\alpha \right)}f\left(x\right)=\frac{1}{A_n^{\left(\gamma \right)}}\sum _{k=0}^n{A}_{n-k}^{\left(\gamma -1\right)}{S}_k^{\left(\alpha \right)}f\left(x\right),\left(n=0,1,2,...\right)$$

$${\Delta }_{x}f\left(t\right)=f\left(x+t\right)-f\left(x\right).$$

$${\int }_0^u{e}^{-t}{t}^{\alpha }\left|{\Delta }_{0}f\left(t\right)\right|dt=o\left(u^{\alpha +1}\right)\text{ as }u\to 0^{+}$$

$${\int }_1^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma -\frac{1}{3}}\left|{\Delta }_{0}f\left(t\right)\right|dt<\infty ,$$

$$\left|S_n^{\left(\gamma ,\alpha \right)}f\left(0\right)-f\left(0\right)\right|=o\left(1\right)\text{ as }n\to \infty .$$

$$\frac{u^{-\left(\alpha +1\right)}}{\Gamma \left(\alpha +1\right)}{\int }_0^u{e}^{-\frac{t}{2}}{t}^{\alpha }\left|{\Delta }_{0}f\left(t\right)\right|dt=O\left(\omega \left(u\right)\right)\left(u>0\right)$$

$$\frac{1}{\Gamma \left(\alpha +1\right)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma -\frac{1}{3}}\left|{\Delta }_{0}f\left(t\right)\right|dt=O\left(\omega \left(1/u\right)\right)\left(u\ge 1\right).$$

$$\left|S_n^{\left(\gamma ,\alpha \right)}f\left(0\right)-f\left(0\right)\right|=O\left(n^{\eta +\frac{2\left(\alpha -\gamma \right)+1}{4}}\right)\sum _{k=1}^n\frac{\omega \left(1/k\right)}{k^{{}^{\eta +\frac{2\left(\alpha -\gamma \right)+1}{4}+1}}}+O\left(\omega \left(1/n^{\eta }\right)\right)$$

$$\left|S_n^{\left(\gamma ,\alpha \right)}f\left(0\right)-f\left(0\right)\right|=o\left(1\right)\text{ as }n\to \infty .$$

$$\left|S_n^{\left(\gamma ,\alpha \right)}f\left(0\right)-f\left(0\right)\right|=O\left(n^{\frac{2\left(\alpha -\gamma \right)+1}{4}}\omega \left(n^{\eta }\right)\right)+O\left(\omega \left(1/n^{\eta }\right)\right)+O\left(\omega \left(1/n\right)\right),$$

$$\left|S_n^{\left(\gamma ,\alpha \right)}f\left(0\right)-f\left(0\right)\right|=O\left(\omega \left(1/n^{\eta }\right)\right),$$

$$|{\Delta }_0{f}_1(t)|=|e^{-\frac{t}{2}}-1|⩽\frac{t}{2}$$

$$\frac{u^{-(\alpha +1)}}{\Gamma (\alpha +1)}{\int }_0^u{e}^{-\frac{t}{2}}{t}^{\alpha }|{\Delta }_0{f}_1(t)|dt$$

$$⩽\frac{u^{-(\alpha +1)}}{2\Gamma (\alpha +1)}{\int }_0^u{t}^{\alpha +1}dt=\frac{1}{2\Gamma (\alpha +1)(\alpha +2)}{\omega }_1(u)$$

$$\frac{1}{{\omega }_1\left(\frac{1}{u}\right)\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma -\frac{1}{3}}|{\Delta }_0{f}_1(t)|dt$$

$$⩽\frac{u}{2\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +\frac{2}{3}}dt⩽\frac{1}{2\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +\frac{5}{3}}dt$$

$$⩽\frac{1}{2\Gamma (\alpha +1)}{\int }_0^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +\frac{5}{3}}dt=\frac{1}{2\Gamma (\alpha +1)}{\int }_0^{\infty }{e}^{-\frac{t}{2}}{t}^{-1+\alpha -\gamma +\frac{8}{3}}dt$$

$$=\frac{1}{\Gamma (\alpha +1)}{2}^{\alpha -\gamma +\frac{5}{3}}\Gamma \left(\alpha -\gamma +\frac{8}{3}\right)<\infty$$

$$|S_n^{(\gamma ,\alpha )}{f}_1(0)-f_1(0)|=O\left(n^{\eta +\frac{2(\alpha -\gamma )+1}{4}}\right)\sum _{k=1}^n\frac{1}{k^{\eta +\frac{2(\alpha -\gamma )+1}{4}+2}}+O\left(\frac{1}{n^{\eta }}\right).$$

$$\frac{u^{-(\alpha +1)}}{\Gamma (\alpha +1)}{\int }_0^u{e}^{-\frac{t}{2}}{t}^{\alpha }|{\Delta }_0{f}_2(t)|dt$$

$$⩽\frac{u^{-(\alpha +1)}}{\Gamma (\alpha +1)}{\int }_0^u{t}^{\alpha +\delta }dt=\frac{1}{\Gamma (\alpha +1)(\alpha +\delta +1)}{\omega }_2(u)$$

$$\frac{1}{{\omega }_2\left(\frac{1}{u}\right)\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma -\frac{1}{3}}|{\Delta }_0{f}_2(t)|dt$$

$$⩽\frac{u^{\delta }}{\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +\delta -\frac{1}{3}}dt⩽\frac{1}{\Gamma (\alpha +1)}{\int }_u^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +2\delta -\frac{1}{3}}dt$$

$$⩽\frac{1}{\Gamma (\alpha +1)}{\int }_0^{\infty }{e}^{-\frac{t}{2}}{t}^{\alpha -\gamma +2\delta -\frac{1}{3}}dt=\frac{1}{\Gamma (\alpha +1)}{\int }_0^{\infty }{e}^{-\frac{t}{2}}{t}^{-1+\alpha -\gamma +2\delta +\frac{2}{3}}dt$$

$$=\frac{1}{\Gamma (\alpha +1)}{2}^{\alpha -\gamma +2\delta +\frac{2}{3}}\Gamma \left(\alpha -\gamma +2\delta +\frac{2}{3}\right)<\infty$$

Full text

**Key words: **Rate of approximation, summability of Fourier–Laguerre series

**2010 Mathematics Subject Classification: **42A24

1 Introduction

Let L_{w}\be the class of all real–valued functions, integrable in the Lebesgue sense over $\mathbb{R}^{+}$ with the weight $w\left(t\right)=e^{-t}t^{\alpha}$ $\left(\alpha>-1\right)$, i.e.

[TABLE]

We will consider the Fourier–Laguerre series

[TABLE]

where

[TABLE]

and

[TABLE]

Let define the $\left(C,\gamma\right)$ - means  of  partial sums

[TABLE]

of $S^{\left(\alpha\right)}f$ as follows

[TABLE]

and let

[TABLE]

The deviation $S_{n}^{\left(\gamma,\alpha\right)}f\left(0\right)-f\left(0\right)$ was examined in the papers [4], [6] and [1] as follows:

Theorem A. [1, Theorem 1] Let $f\in L_{w},$ $\alpha>-1$ and $\gamma>\alpha+\frac{1}{2}.$ If a function $f$ satisfies the conditions

[TABLE]

and

[TABLE]

then

[TABLE]

Similar results in a case of norm approximation due of C. Markett and E. L. Poiani in papers [2] and [3] were obtained.

We will say that a nonnegative function $\omega$ is a function of the modulus of continuity type if it is nondecreasing continuous function on $[0,\infty)$ having the following conditions: $\omega\left(0\right)=0$ and $\omega\left(\delta_{1}+\delta_{2}\right)\leq\omega\left(\delta_{1}\right)+\omega\left(\delta_{2}\right)$ for any $\delta_{1},\delta_{2}\in[0,\infty)$.

In this paper, we will study the upper bound of the quantity $\left|S_{n}^{\left(\gamma,\alpha\right)}f\left(0\right)-f\left(0\right)\right|$ by some means of a function of the modulus of continuity type $\omega.$ From our result we will derive some corollaries, remark and construct some examples.

2 Statement of the results

First we present the estimate of the quantity $\left|S_{n}^{\left(\gamma,\alpha\right)}f\left(0\right)-f\left(0\right)\right|$.

Theorem 1**.**

Let $f\in L_{w},$ $\alpha>-1$, $\gamma>\alpha+\frac{1}{2}$ and let a function $\omega$ of the modulus of continuity type satisfy the conditions:

[TABLE]

and

[TABLE]

Then

[TABLE]

for $0<\eta<-\frac{2\left(\alpha-\gamma\right)+1}{4}.$

Now, we formulate some corollaries and remark.

Corollary 1**.**

Under the assumptions of the above theorem

[TABLE]

Remark 1**.**

Using Theorem 1 and Corollary 1 we obtain the result of D. P. Gupta from Theorem A.

Corollary 2**.**

Analyzing the proof of Theorem 1 we can obtain, under the assumptions of this theorem, the following more precise estimate

[TABLE]

when $\frac{2\left(\alpha-\gamma\right)+1}{4}+1<0$.

In the special case, taking $\eta=-\frac{2\left(\alpha-\gamma\right)+1}{8}$ , we have

[TABLE]

when $\eta\leq 1$.

3 Examples

Let $f_{1}(t)=e^{-\frac{t}{2}}$ and $\omega_{1}(t)=t$ for $t\geqslant 0$.

It is clear that $f_{1}\in L_{w}$. Moreover, applying the Lagrange mean value theorem we get that

[TABLE]

for $t\geq 0$. Therefore, by elementary calculations we get

[TABLE]

[TABLE]

for $u>0$ and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for $u\geq 1$ and $\alpha-\gamma+\frac{8}{3}>0$.

Hence the function $f_{1}$ satisfies the conditions $(1)$ and $(2)$. Using Theorem 1 we get the following estimate for $|S_{n}^{(\gamma,\alpha)}f_{1}(0)-f_{1}(0)|$:

Example 1**.**

Let $\alpha>-1,$ $\alpha+\frac{1}{2}<\gamma<$ $\alpha+\frac{8}{3}$ and $0<\eta<-\frac{2(\alpha-\gamma)+1}{4}$. Then

[TABLE]

Suppose $f_{2}\left(t\right)=t^{\delta}$ and $\omega_{2}\left(t\right)=t^{\delta}$ for $\delta\in(0,1]$ and $t\geqslant 0$.

Obviously $f_{2}\in L_{w}$. In addition, it is easy to show that

[TABLE]

[TABLE]

for $u>0$ and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for $u\geq 1$ and $\alpha-\gamma+2\delta+\frac{2}{3}>0$.

Therefore the function $f_{2}$ satisfies the conditions $(1)$ and $(2)$. Using Theorem 1 we get the following estimate for $|S_{n}^{(\gamma,\alpha)}f_{2}(0)-f_{2}(0)|$:

Example 2**.**

Let $\alpha>-1,$ $\delta\in(0,1]$, $\alpha+\frac{1}{2}<\gamma<\alpha+2\delta+\frac{2}{3}$ and $0<\eta<-\frac{2(\alpha-\gamma)+1}{4}$. Then

[TABLE]

4 Auxiliary results

We begin this section by some notations from [5] . We have

[TABLE]

and therefore

[TABLE]

Hence, by evidence equality

[TABLE]

we have

[TABLE]

Next, we present the useful estimates:

Lemma 1**.**

[5, p. 172]** Let $\beta$ be an arbitrary real number, $c$ and $\delta$ be fixed positive constants. Then

[TABLE]

Lemma 2**.**

[5, p. 235]** Let $\beta$ and $\lambda$ be arbitrary real numbers, $\delta>0$ and $0<\theta<4$ . Then

[TABLE]

Lemma 3**.**

[7, Vol. I, (1.15) and Theorem 1.17]** If $\gamma>-1,$ then

[TABLE]

and $A_{n}^{\left(\gamma\right)}$ is positive for $\gamma>-1$ increasing (as a function of $n$) for $\gamma>0$ and decreasing for $-1<\gamma<0.$

5 Proofs of Theorems

5.1 Proofs of Theorem 1

It is clear that if

[TABLE]

then

[TABLE]

By Lemma 1, Lemma 3 and (1)

[TABLE]

with $0<\eta<-\frac{2\left(\alpha-\gamma\right)+1}{4}$.

Using Lemma 1, we get

[TABLE]

Let $F_{\alpha}\left(y\right)=\frac{y^{-\left(\alpha+1\right)}}{\Gamma\left(\alpha+1\right)}\int_{0}^{y}e^{-\frac{u}{2}}u^{\alpha}\left|\Delta_{0}f\left(u\right)\right|du$. Applying Lemma 3 and integrating by parts with $\gamma>\alpha+\frac{1}{2}$ and $\alpha>-1$ we have

[TABLE]

[TABLE]

[TABLE]

By (1) we obtain

[TABLE]

with $0<\eta<-\frac{2\left(\alpha-\gamma\right)+1}{4}.$

Applying Lemma 2  with $\alpha+\gamma+1$ instead of $\beta$ and $\lambda=\frac{2\alpha+2\gamma+3}{4}$ (since $\max\left(\lambda-\frac{1}{2},\frac{\alpha+\gamma+1}{2}-\frac{1}{4}\right)=\frac{2\alpha+2\gamma+1}{4}$) we have

[TABLE]

Using Lemma 3 and integrating by parts with $\gamma>\alpha+\frac{1}{2}$ and $\alpha>-1$, we get

[TABLE]

[TABLE]

[TABLE]

By (1) we obtain

[TABLE]

[TABLE]

[TABLE]

with $0<\eta<-\frac{2\left(\alpha-\gamma\right)+1}{4}.$

If $\lambda=\gamma+\frac{1}{3}$ then $\lambda-\frac{1}{3}=\gamma>\frac{\alpha+\gamma+1}{2}-\frac{1}{4}$ since $\gamma>\alpha+\frac{1}{2}$. So, applying Lemma 2 with $\alpha+\gamma+1$ instead of $\beta$ and $\lambda=\gamma+\frac{1}{3}$ we obtain

[TABLE]

Next, using Lemma 3 and (2) we get

[TABLE]

Finally, collecting the above estimates we have

[TABLE]

and our proof is completed. $\blacksquare$

6 Conclusions

We investigated pointwise approximation of real–valued functions, integrable in the Lebesgue sense over $\mathbb{R}^{+}$ with the weight $w\left(t\right)=e^{-t}t^{\alpha}$ $\left(\alpha>-1\right)$ by the $\left(C,\gamma\right)$ - means  of  partial sums of their Fourier–Laguerre series. In particular, we estimated the deviation $\left|S_{n}^{\left(\gamma,\alpha\right)}f\left(0\right)-f\left(0\right)\right|$ by means of a function of the modulus of continuity type $\omega$. From our result some corollaries were derived and some examples were constructed.