Further Spectral Properties of the Weighted Finite Transform Operator and Approximation in Weighted Sobolev Spaces

TL;DR

This paper investigates the spectral properties of the weighted finite Fourier transform operator, focusing on the decay of singular values, bounds of GPSWFs, and their approximation capabilities in weighted Sobolev spaces, supported by numerical examples.

Contribution

It provides new insights into the decay rate of singular values and approximation quality of GPSWFs in weighted Sobolev spaces, with detailed bounds and numerical illustrations.

Findings

Singular values decay super-exponentially.

GPSWFs have specific local bounds.

Effective approximation in weighted Sobolev spaces.

Abstract

In this work, we first give some mathematical preliminairies concerning the generelized prolate spheroidal wave functions(GPSWFs). This set of special functions have been introduced in [21]and [13] and they are defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space.Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Figures

Tables

Table 1. Table 1: Values of the approximation of Weierstrass-Mandelbrot function ‖ M s , 2 − S N ( α ) ​ ( M s , 2 ) L 2 ​ ( I , ω α ) ‖ norm subscript 𝑀 𝑠 2 superscript subscript 𝑆 𝑁 𝛼 subscript subscript 𝑀 𝑠 2 superscript 𝐿 2 𝐼 subscript 𝜔 𝛼 {\left\|{M_{s,2}-S_{N}^{(\alpha)}(M_{s,2})_{L^{2}(I,\omega_{\alpha})}}\right\|} for different values of s 𝑠 s and α 𝛼 \alpha

$\alpha$	$s=0.25$	$s=0.5$	$s=0.75$	$s=1$
$0.1$	$1.69146E-04$	$5.42800E-04$	$1.74173E-03$	$5.61554E-0.3$
$0.5$	$1.90589E-04$	$6.07253E-04$	$1.93120E-03$	$6.15556E-0.3$
$1$	$2.12572E-04$	$6.72661E-04$	$2.12113E-03$	$6.68912E-0.3$
$1.5$	$2.30518E-04$	$7.25472E-04$	$2.27216E-03$	$7.10411E-03$
$2.0$	$2.45810E-04$	$7.70063E-04$	$2.39797E-03$	$7.44278E-03$

Equations

$$\text{mbox}Findg=\arg \underset{f\in B_c}{\max }\frac{{\int }_{-1}^1|f(t){|}^{2}dt}{{\int }_{\mathbb{R}}|f(t){|}^{2}dt}$$

$$B_c=\{f\in L^2(\mathbb{R}),\text{mbox}Support\hat{f}\subseteq [-c,c]\}.$$

$${\mathcal{Q}}_{c}f(x)={\int }_{-1}^1\frac{\sin (c(x-y))}{\pi (x-y)}f(y)dy$$

$${\mathcal{L}}_{c}f(x)=(1-x^2)f^{\prime \prime }(x)-2x{f}^{\prime }(x)-c^2{x}^{2}f(x)$$

$${\mathcal{L}}_c^{(\alpha )}\phi (x)=(1-x^2){\phi }^{\prime \prime }(x)-2\left(\alpha +1\right)x{\phi }^{\prime }(x)-c^2{x}^2\phi (x)={\mathcal{L}}_0\phi (x)-c^2{x}^2\phi (x),\text{mbox}where\alpha >-1;c>0,$$

$${\mathcal{F}}_c^{(\alpha )}f(x)={\int }_{-1}^1{e}^{icxy}f(y)(1-y^2{)}^{\alpha }dy,\alpha >-1.$$

$$H_{\alpha }^r(I)=\{f\in L_{I,\omega }^2:{\Vert f\Vert }_{\alpha ,r}<\infty \}\text{mbox}where{\Vert f\Vert }_{\alpha ,r}^2=\sum _{k=0}^r{\Vert f^{(k)}\Vert }_{\omega }^2$$

$${\Vert f\Vert }_{\alpha ,r}=\sum _{k=0}^r{\Vert f^{(k)}\Vert }_{\omega }^2\text{mbox}and|f{|}_{\alpha ,r}=\Vert f^{(k)}{\Vert }_{{\omega }_{\alpha }}$$

$${\omega }_{\alpha }(t)=(1-t^2{)}^{\alpha }\text{mbox}and{\Vert .\Vert }_{\omega }={\Vert .\Vert }_{L^2(I,{\omega }_{\alpha })}$$

$$\Vert f-S_N(f){\Vert }_{L^2(I,{\omega }_{\alpha })}$$

$$|{\mu }_n^{(\alpha )}(c)|\le \frac{k_{\alpha }}{c^{\frac{\alpha }{2}+1}\log \left(\frac{2n-1}{ec}\right)}{\left(\frac{ec}{2n-1}\right)}^{n+\frac{\alpha }{2}},k_{\alpha }={\left(\frac{2}{e}\right)}^{1+\frac{\alpha }{2}}\pi (\Gamma (\alpha +1){)}^{1/2}.$$

$$\lambda_{n}^{(\alpha)}(c)\leq\frac{K_{\alpha}}{c^{\alpha+1}\log^{2}(\frac{2n-1}{ec})}\Big{(}\frac{ec}{2n-1}\Big{)}^{2n+\alpha}\quad\mbox{with}\quad K_{\alpha}=\frac{\pi}{2}\big{(}2/e\big{)}^{\alpha+2}\Gamma(\alpha+1)$$

$${\mathcal{F}}_c^{(\alpha )}f(x)={\int }_{-1}^1{e}^{icxy}f(y){\omega }_{\alpha }(y)dy,{\omega }_{\alpha }(y)=(1-y^2{)}^{\alpha },\alpha >0$$

$${\mathcal{Q}}_c^{(\alpha )}g(x)={\int }_{-1}^1\frac{c}{2\pi }{\mathcal{K}}_{\alpha }(c(x-y))g(y){\omega }_{\alpha }(y)dy,{\mathcal{K}}_{\alpha }(x)=\sqrt{\pi }{2}^{\alpha +1/2}\Gamma (\alpha +1)\frac{J_{\alpha +1/2}(x)}{x^{\alpha +1/2}}.$$

$${\mathcal{L}}_c^{(\alpha )}(f)(x)=-\frac{1}{{\omega }_{\alpha }(x)}\frac{d}{dx}\left[{\omega }_{\alpha }(x)(1-x^2)f^{\prime }(x)\right]+c^2{x}^{2}f(x),{\omega }_{\alpha }(x)=(1-x^2{)}^{\alpha }.$$

$$n(n+2\alpha +1)\le {\chi }_n^{\alpha }(c)\le n(n+2\alpha +1)+c^2,\forall n\ge 0.$$

$$B_{c}^{(\alpha)}=\{f\in L^{2}(\mathbb{R}),\,\,\mbox{Support }\widehat{f}\subseteq[-c,c],\,\,\widehat{f}\in L^{2}\big{(}(-c,c),\omega_{-\alpha}(\frac{\cdot}{c})\big{)}\}.$$

$$\|f\|^{2}_{L^{2}\big{(}(-c,c),\omega_{-\alpha}(\frac{\cdot}{c})\big{)}}=\int_{-c}^{c}|f(t)|^{2}\omega_{-\alpha}\left(\frac{t}{c}\right)\,dt.$$

$${\lambda }_n^{(\alpha )}(c)<e^{-2n\log (\frac{bn}{c})}\forall n\ge N_b,$$

$${\lambda }_n^{(\alpha )}(c)\le C_{\alpha }\exp \left(-(2n+1)\left[\log \left(\frac{4n+4\alpha +2}{ec}\right)+C_{\alpha }\frac{c^2}{2n+1}\right]\right),\forall n\ge N_{\alpha }(c).$$

$${\psi }_{n,c}^{(\alpha )}(x)=\sum _{k\ge 0}{\beta }_k^n{\tilde{P}}_k^{(\alpha ,\alpha )}(x),x\in [-1,1].$$

$${\tilde{P}}_k^{(\alpha ,\alpha )}(x)=\frac{1}{\sqrt{h_k}}{P}_k^{(\alpha ,\alpha )}(x),h_k=\frac{2^{2\alpha +1}{\Gamma }^2(k+\alpha +1)}{k!(2k+2\alpha +1)\Gamma (k+2\alpha +1)}.$$

$$\displaystyle\frac{\sqrt{(k+1)(k+2)(k+2\alpha+1)(k+2\alpha+2)}}{(2k+2\alpha+3)\sqrt{(2k+2\alpha+5)(2k+2\alpha+1)}}c^{2}\beta_{k+2}^{n}+\big{(}k(k+2\alpha+1)+c^{2}\frac{2k(k+2\alpha+1)+2\alpha-1}{(2k+2\alpha+3)(2k+2\alpha-1)}\big{)}\beta_{k}^{n}$$

$$H_{\alpha }^r(I)=\{f\in L^2(I,{\omega }_{\alpha }):{\Vert f\Vert }_{\alpha ,r}<\infty \}\text{mbox}where{\Vert f\Vert }_{\alpha ,r}=\sum _{k=0}^r{\Vert f^{(k)}\Vert }_{\omega }^2$$

$$|{\mu }_n^{(\alpha )}(c)|\le \frac{k_{\alpha }}{c^{\frac{\alpha }{2}+1}\log \left(\frac{2n-1}{ec}\right)}{\left(\frac{ec}{2n-1}\right)}^{n+\frac{\alpha }{2}},k_{\alpha }={\left(\frac{2}{e}\right)}^{1+\frac{\alpha }{2}}\pi (\Gamma (\alpha +1){)}^{1/2}.$$

$$\lambda_{n}^{(\alpha)}(c)\leq\frac{K_{\alpha}}{c^{\alpha+1}\log^{2}(\frac{2n-1}{ec})}\Big{(}\frac{ec}{2n-1}\Big{)}^{2n+\alpha}\quad\mbox{with}\quad K_{\alpha}=\frac{\pi}{2}\big{(}2/e\big{)}^{\alpha+2}\Gamma(\alpha+1)$$

$${\lambda }_n=\underset{f\in S_n}{\min }\underset{f\in S_n^{\perp },\Vert f{\Vert }_{\mathcal{G}}=1}{\max }<Tf,f{>}_{\mathcal{G}},$$

$$S_n=\text{mbox}Span\left\{{\tilde{P}}_0^{(\alpha ,\alpha )},{\tilde{P}}_1^{(\alpha ,\alpha )},\dots ,{\tilde{P}}_{n-1}^{(\alpha ,\alpha )}\right\}$$

$$f=\sum _{k\ge n}{a}_k{\tilde{P}}_k^{(\alpha ,\alpha )}\in S_n^{\perp },\parallel f{\parallel }_{L_{(I,{\omega }_{\alpha })}^2}=\sum _{k\ge n}|a_k{|}^2=1.$$

$$\Vert {\mathcal{F}}_c^{(\alpha )}{P}_k^{(\alpha ,\alpha )}{\Vert }_{L^2(I,{\omega }_{\alpha })}=\parallel \sqrt{\pi }{\left(\frac{2}{cx}\right)}^{\alpha +1/2}\frac{\Gamma (k+\alpha +1)}{\Gamma (k+1)}{J}_{k+\alpha +1/2}(cx){\parallel }_{L^2(I,{\omega }_{\alpha })}.$$

Full text

**Further Spectral Properties of the Weighted Finite Fourier Transform Operator and Approximation in Weighted Sobolev Spaces.

**NourElHouda Bourguibaa and Ahmed Souabnia 111 Corresponding author: Ahmed Souabni, email: souabniahmed@yahoo.fr

This work was supported in part by the Tunisian DGRST research grants UR 13ES47.

a University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Tunisia.

Abstract— In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWFs). This set of special functions have been introduced in [21] and [13] and they are defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space.Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

1 Introduction

We first recall that for $c>0$, the classical prolate spheroidal wave functions (PSWFs) were first discovered and studied by D. Slepian and his co-authors, see [18, 19]. These PSWFs are defined as the solutions of the following energy maximization problem :

[TABLE]

Where $B_{c}$ is the classical Paley-Winer space, defined by

[TABLE]

Here, $\widehat{f}$ is the Fourier transform of $f\in L^{2}(\mathbb{R}),$ defined by ${\displaystyle\widehat{f}(\xi)=\lim_{A\rightarrow+\infty}\int_{[-A,A]}e^{-ix\xi}f(x)\,dx.}$ Moreover, it has been shown in [18], that the PSWFs are also the eigenfunctions of the integral operator ${\displaystyle\mathcal{Q}_{c}}$ defined on $L^{2}(-1,1)$ by

[TABLE]

as well as the eigenfunctions of a commuting differential operator $\mathcal{L}_{c}$ with $\mathcal{Q}_{c}$ and given by

[TABLE]

In this paper, we are interested in a weighted family of PSWFs called the generalized prolate spheroidal wave functions (GPSWFs), recently given in [13] and[21] . They are defined as the eigenfunctions of the Gegenbauer perturbed differential operator,

[TABLE]

as well as the eigenfunctions of the following commuting weighted finite Fourier transform integral operator,

[TABLE]

The question of quality of approximation by PSWFs has attracted some interests. In fact the authors in [10] have given an estimate of the decay of the coefficients of the PSWFs series expansion of $f\in H^{s}(I)$. Here $H^{s}(I)$ is the Sobolev space over $I=[-1,1]$ and of exponent $s>0$. Later in [21] the authors studied the convergence of the expansion of functions $f\in H^{s}(I)$ in a basis of PSWFs. We should mention that the problem of the best choice of the value of the bandwidth $c>0$ arises here. Numerical answers were given in [21]. Recently in [7], the authors have given a precise answer to the previous problem. In this work, we further study the convergence rate of the projection ${\displaystyle S_{N}\cdot f=\displaystyle\sum_{k=0}^{N}<f,\psi_{n,c}^{(\alpha)}>\psi_{n,c}^{(\alpha)}}$ to $f$ . Here $\psi_{n,c}^{(\alpha)}$ are the GPSWFs and $f\in H^{s}_{\alpha}(I),$ a weighted Sobolev space defined by:

[TABLE]

equipped with the norm and semi-norm

[TABLE]

Here

[TABLE]

In this work , we will give the following result

Theorem 1**.**

Let $c>1$ and $\alpha>0$, then there exist constants $K>0$ and $a>0$ such that, when $N>m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ and $f\in H^{s}_{\alpha}$; $s>0$, we have the inequality

[TABLE]

Where $S_{N}(f)(t)=\displaystyle{\sum_{n<N}}<f,\psi_{n,c}^{\alpha}>_{L^{2}_{\alpha}(I)}\psi_{n,c}^{\alpha}$

Note that the subject of spectral approximation in weighred Sobolev spaces has been studied by Canuto and Quarteroni [9] in the Legendre and Chebyshev case, Bernardi and Maday [3]have developed Geganbauer approximations in the same context. Later in [12] Guo and Wang developed the approximation result in the Jacobi case. Our study of the convergence rate of ${\left\|{S_{N}-Id}\right\|}$ is done by using an estimate for the decay of the eigenvalues $\lambda^{(\alpha)}_{n}(c)$ of the self-adjoint operator ${\displaystyle\mathcal{Q}_{c}^{(\alpha)}=\frac{c}{2\pi}\mathcal{F}_{c}^{(\alpha)^{*}}\mathcal{F}_{c}^{(\alpha)},}$ that is $\lambda_{n}^{(\alpha)}(c)=\frac{c}{2\pi}{|\mu_{n}^{(\alpha)}(c)|^{2}}$, where $\mu_{n}^{(\alpha)}$ is the n-th eigenvalues of the integral operator given by (2). That’s why the first part of this work will be devoted to the study of the spectral decay of the eigenvalues of the operator ${\displaystyle\mathcal{Q}_{c}^{(\alpha)}}$.

Nonetheless, the decay rate of the eigenvalues of the integral operator has been studied in [14] where a first asymptotically decay rate of $(\lambda_{n}^{(\alpha)})_{n}$ for $0<\alpha<3/2$ has been given . In this work we further improve this last result by giving clearly simpler and more general proof. We will prove that

Theorem 2**.**

For given real numbers $c>0$,$\;\;\;\alpha\geq 0$ and for any integer $n>\frac{ec+1}{2},$ we have

[TABLE]

and

[TABLE]

This work is organized as follows. In section 2, we give some mathematical preliminaries concerning the GPSWFs. In section 3, we give some further estimates of the GPSWFs and some estimates of the eigenvalues of both integral and differential operators associated with GPSWFs.In section 4, we study,the quality of approximation by the GPSWFs of functions from a weighted Sobolev space. Finally, in section 5 we give some numerical examples that illustrate the different results of this work.

2 Mathematical Preliminaries

In this section, we give some mathematical preliminaries concerning some properties as well as the computation of the GPSWFs. These mathematical preliminaries are used to describe the different results of this work. The GPSWFs are defined as the eigenfunctions of the weighted finite Fourier transform operator $\mathcal{F}_{c}^{(\alpha)},$ defined by

[TABLE]

They are also the eigenfunctions of the operator ${\displaystyle\mathcal{Q}_{c}^{(\alpha)}=\frac{c}{2\pi}\mathcal{F}_{c}^{({\alpha})^{*}}\circ\mathcal{F}_{c}^{(\alpha)}},$ defined on $L^{2}{(I,\omega_{\alpha})}$ by

[TABLE]

Here $J_{\alpha}(\cdot)$ is the Bessel function of the first kind and order $\alpha>-1.$ Moreover, the eigenvalues $\mu_{n}^{(\alpha)}(c)$ and $\lambda_{n}^{(\alpha)}(c)$ of $\mathcal{F}_{c}^{(\alpha)}$ and $\mathcal{Q}_{c}^{(\alpha)}$ are related to each others by the identity ${\displaystyle\lambda_{n}^{(\alpha)}(c)=\frac{c}{2\pi}|\mu_{n}^{(\alpha)}(c)|^{2}}.$ Note that the previous two integral operators commute with the following Gegenbauer-type Sturm-Liouville operator $\mathcal{L}_{c}^{(\alpha)},$ defined by

[TABLE]

Also, note that the $(n+1)-$th eigenvalue $\chi_{n}^{\alpha}(c)$ of $\mathcal{L}_{c}^{(\alpha)}$ satisfies the following classical inequalities,

[TABLE]

For more details, see [13]. We will denote by$(\psi_{n,c}^{(\alpha)})_{n\geq 0},$ the set of the eigenfunctions of $\mathcal{F}_{c}^{(\alpha)},\mathcal{Q}_{c}^{(\alpha)}$ and $\mathcal{L}_{c}^{(\alpha)}.$ They are called generalized prolate spheroidal wave functions (GPSWFs). It has been shown that $\{\psi_{n,c}^{(\alpha)},n\geq 0\}$ is an orthogonal basis of $L^{2}(I,\omega_{\alpha}),I=[-1,1].$ We recall that the restricted Paley-Wiener space of weighted $c-$band-limited functions has been defined in [13] by

[TABLE]

Here, L^{2}\big{(}(-c,c),\omega_{-\alpha}(\frac{\cdot}{c})\big{)} is the weighted $L^{2}(-c,c)-$space with norm given by

[TABLE]

It has been shown in [13] that the eigenvalues $\lambda_{n}^{(\alpha)}(c)$ decay asymptotically at a super-exponential rate that is for $0<b<4/e$ there exists $N_{b}\in\mathbb{N}$ such that :

[TABLE]

Note that a second decay rate of the $\lambda_{n}^{(\alpha)}(c)$ and valid for $0<\alpha<\frac{3}{2},$ has been recently given in [14]. More precisely, it has been shown in [14], that if $c>0$ and $0<\alpha<\frac{3}{2}$, then there exist $N_{\alpha}(c)\in\mathbb{N}$ and a constant $C_{\alpha}>0$ such that

[TABLE]

In [13], the authors have proposed the following scheme for the computation of the GPSWFs. In fact, since $\psi^{(\alpha)}_{n,c}\in L^{2}(I,\omega_{\alpha}),$ then its series expansion with respect to the Jacobi polynomials basis is given by

[TABLE]

Here, $\widetilde{P}^{(\alpha,\alpha)}_{k}$ denotes the normalized Jacobi polynomial of degree $k,$ given by

[TABLE]

The expansion coefficients $\beta_{k}^{n}$ as well as the eigenvalues $\chi_{n}^{\alpha}(c)$ this system is given by

[TABLE]

We denote, for any $r\in\mathbb{N}$, $H^{r}_{\alpha}(I)$ the Jacobi-weighted Sobolev space by :

[TABLE]

In particular, $L^{2}(I,\omega_{\alpha})=H^{0}_{\alpha}(I)$ and ${\left\|{.}\right\|}_{\alpha,0}={\left\|{.}\right\|}_{\omega}$.

For any real r, we define the weighted Sobolev space $H^{r}_{\alpha}$ by interpolation H^{r}_{\alpha}=\Big{[}H^{[r]}_{\alpha},H^{[r]+1}_{\alpha}\Big{]} as it is done in [2] and [3] We denote by $\mathcal{C}_{\infty}(I)$ the set of infinitely differentiable functions defined on $I$. Let $H_{0}^{s}(I)$ be the completion of $\mathcal{C}_{\infty}(I)$ in $H^{s}(I)$. $H_{0}^{s}(I)$ is a Hilbert space under the inner product. For $s\geq 0$, we define $H_{0}^{s}(I)$by interpolation as in [2] and [3].

3 Further Estimates of the GPSWFs and their associated eigenvalues.

In this section, we first give some new estimates of the eigenvalues $\mu_{n}^{\alpha}(c)$ associated with the GPSWFs. Then, we study a local estimate of these GPSWFs.

3.1 Estimates of the eigenvalues.

We first prove in a fairly simple way, the super-exponential decay rate of the eigenvalues of the weighted Fourier transform operator. We should mention that the techniques used in this proof is inspired from the recent paper [8], where a similar result has been given in the special case $\alpha=0$.

Theorem 3**.**

For given real numbers $c>0$,$\;\;\;\alpha\geq 0$ and for any integer $n>\frac{ec+1}{2},$ we have

[TABLE]

and

[TABLE]

Proof: We first recall the Courant-Fischer-Weyl Min-Max variational principle concerning the eigenvalues of a compact self-adjoint operator $T$ on a Hilbert space $\mathcal{G},$ with positive eigenvalues arranged in the decreasing order $\lambda_{0}\geq\lambda_{1}\geq\cdots\geq\lambda_{n}\geq\cdots,$ then we have

[TABLE]

where $S_{n}$ is a subspace of $\mathcal{G}$ of dimension $n.$ In our case, we have $T=\mathcal{F}_{c}^{\alpha^{*}}\mathcal{F}_{c}^{(\alpha)},$ $\mathcal{G}=L^{2}(I,\omega_{\alpha}).$ We consider the special case of

[TABLE]

and

[TABLE]

Here the $\widetilde{P}^{(\alpha,\alpha)}_{k}$ are the normalized Jacobi polynomials given by (12). From [[17] page 456], we have

[TABLE]

and

[TABLE]

By using the well-known bound of the Bessel function given by

[TABLE]

one gets,

[TABLE]

Next, we use (12) and the following useful inequalities for the Gamma function, see [22]

[TABLE]

Then, we have

[TABLE]

Since the function $g_{a}(x)=\left(1+\frac{a}{x}\right)^{a+x}$ for $x\geq 1$ and $a>-1$ is decreasing on $[1,+\infty[$. Then, we have

[TABLE]

Hence, For the previous $f\in S_{n}^{\perp},$ and by using Hölder’s inequality, combined with the Minkowski’s inequality for an infinite sums, and taking into account that $\|f\|_{L^{2}(I,\omega_{\alpha})}=1,$ so that $|a_{k}|\leq 1,$ for $k\geq n,$ one gets:

[TABLE]

The decay of the sequence appearing in the previous sum, allows us to compare this later with its integral counterpart, that is

[TABLE]

Hence, by using (24) and (25), one concludes that

[TABLE]

To conclude the proof of the theorem, it suffices to use the Courant-Fischer-Weyl Min-Max variational principle.$\qquad\Box$

Remark 1**.**

The decay rate given by the last proposition improve the two results mentioned above (9) and (10). Indeed, we have a more precise result furthermore the proof given in [14] contains two serious drawbacks. It is only valid in the particular case $0<\alpha<3/2$ and clearly more sophisticated than the previous one.

3.2 Local estimates of the GPSWFs

In this paragraph, we give a precise local estimate of the GPSWFs, which is valid for $0\leq\alpha\leq 1/4.$ Then, this local estimate will be used to provide us with a new lower bound for the eigenvalues $\chi_{n}^{\alpha}(c)$ of the differential operator $\mathcal{L}_{c}^{\alpha},$ for $0\leq\alpha\leq 1/4.$ For this purpose, we first recall that $\psi_{n,c}^{(\alpha)}$ are the bounded solutions of the following ODE :

[TABLE]

As it is done in [5], we consider the incomplete elliptic integral

[TABLE]

Then, we write $\psi_{n,c}^{(\alpha)}$ into the form

[TABLE]

By combining (27), (29) and using straightforward computations, it can be easily checked that $V(\cdot)$ satisfies the following second order differential equation

[TABLE]

with

[TABLE]

Define $Q_{\alpha}(x)=w_{\alpha}(x)^{2}(1-x^{2})(1-qx^{2}),$ then we have ${\displaystyle\frac{\phi_{\alpha}^{\prime}(x)}{\phi_{\alpha}(x)}=-1/4\frac{Q_{\alpha}^{\prime}(x)}{Q_{\alpha}(x)}.}$ It follows that $\theta_{\alpha}(s)$ can be written as

[TABLE]

Since $Q_{\alpha}(x)=w_{\alpha}^{2}(x)Q_{0}(x),$ then we have ${\displaystyle\frac{Q_{\alpha}^{\prime}(x)}{Q_{\alpha}(x)}=2\frac{w_{\alpha}^{\prime}(x)}{w_{\alpha}(x)}+\frac{Q_{0}^{\prime}(x)}{Q_{0}(x)}}$ and ${\displaystyle\frac{w_{\alpha}^{\prime}(x)}{w_{\alpha}(x)}=-\frac{2\alpha x}{1-x^{2}}.}$ Hence, we have

[TABLE]

The previous equality allows us to prove the following lemma.

Lemma 1**.**

*For any $0\leq\alpha\leq\frac{1}{4}$ and $0<q=c^{2}/\chi_{n}^{\alpha}(c)<\frac{3}{17}$, we have $\theta_{\alpha}(S(x))$ is increasing on $[0.1]$.

Proof: We use the notation $u=1-x^{2}$ and with straightforward computations, we have

[TABLE]

where

[TABLE]

with $G(u)\geq\frac{(1-q)^{2}}{4}(4-4q+u(17q-3))\geq 0,\;\;\forall q>0,\;\;\alpha>0$ then

[TABLE]

Then if $0\leq q\leq\frac{3}{17}$ and $\alpha\leq\frac{1}{4}$ we have

[TABLE]

$\qquad\Box$ As a consequence of the previous lemma, one gets.

Lemma 2**.**

For any $0\leq\alpha\leq\frac{1}{4}$ and for any integer $n\in\mathbb{N},$ with $0<q=\frac{c^{2}}{\chi_{n}^{\alpha}(c)}<\frac{3}{17}$, we have

[TABLE]

Proof.

For $0\leq\alpha\leq\frac{1}{4}$ and $0<q<\frac{3}{17}$ ,we have $\theta_{\alpha}\circ S$ is increasing. Since $S$ is decreasing then $\theta$ is also decreasing. We define $K(s)=|V(s)|^{2}+\frac{1}{\chi_{n}^{\alpha}(c)+\theta(s)}|V^{\prime}(s)|^{2}$ By the fact that $K(s)$ and $\frac{1}{\chi_{n}^{\alpha}(c)+\theta(s)}$ has the same monotonicity and $\theta$ is decreasing , we conclude that $K$ is increasing. Then

[TABLE]

We remark easily that $V(S(0))=\psi_{n,c}^{(\alpha)}(0)$ and $V^{\prime}(S(0))={\psi_{n,c}^{(\alpha)}}^{\prime}(0)$ then we conclude for the first inequality of (33). The second inequality is given later on remark 2 ∎

Next, we give some lower and upper bounds for the eigenvalues $\chi_{n}^{\alpha}(c),$ the $n+1$th eigenvalues of the differential operator. This is given by the following proposition.

Proposition 1**.**

For c and n such that $0\leq\alpha\leq\frac{1}{4}$ and $0<q<\frac{3}{17}$, we have :

[TABLE]

Where

[TABLE]

Proof.

By differentiating with respect to $c$ the differential equation satisfied by GPSWFs , one gets

[TABLE]

Hence, we have

[TABLE]

It’s well known that $\mathcal{L}^{(\alpha)}_{c}$ is a self-adjoint operator and have $\psi_{n,c}^{(\alpha)}$ as eigenfunctions . Hence, we have

[TABLE]

It follows that

[TABLE]

From the fact that ${\left\|{\psi_{n,c}^{(\alpha)}}\right\|}_{L^{2}(I,\omega_{\alpha})}=1$, one gets

[TABLE]

As it is done in [4], we denote by A=\Bigg{[}\psi_{n,c}^{(\alpha)}(0)+\frac{{\psi_{n,c}^{(\alpha)}}^{\prime}(0)}{\chi_{n}^{\alpha}(c)}\Bigg{]}^{1/2} and consider the auxiliary function

[TABLE]

Straightforward computations give us

[TABLE]

with $H(t)\geq 0$ for $t\in[0,1]$. Hence

[TABLE]

So one has the inequality

[TABLE]

That is

[TABLE]

Remark that (33) implies that

[TABLE]

By combining (37) and (38) we conclude that $B^{1/2}$ is bounded below by the largest solution of the equation $X^{2}+2(2\alpha+1)X-1=0$

∎

Remark 2**.**

By using (37) and since $B\leq 1$, one concludes that the constant $A,$ given in (33) satisfies $A^{2}\leq 2\alpha+1.$

4 Approximation by the GPSWFs in Weighted Sobolev spaces

In this section, we study the issue of the quality of spectral approximation of a function $f\in H^{s}_{\alpha}(I)$ by its truncated GPSWFs series expansion.

Note that a different spectral approximation result by the GPSWFs has been already given in [21]. It is important to mention here that the spectral approximation given in [21] is done in different approach. More precisely, by considering the weighted Sobolev space associated with the differential operator defined by

[TABLE]

where $f_{k}$ are expansions coefficients of $f$ in the GPSWFs’s basis. Then it has been shown that:

for any $f\in\widetilde{H}^{r}_{\omega_{\alpha}}(I)$ with $r\geq 0$

[TABLE]

For more details, we refer the reader to [21].

Remark 3**.**

We have the following norm’s comparison given in [21] :

[TABLE]

Note that for large value of $N$ , we have the same decay rate but when $N^{2}$ and $1+c^{2}$ are comparable we can notice clearly the importance of the following result.

Theorem 4**.**

For $\alpha\geq 0$ . For any real number $m\geq 0$, $f\in H^{m}_{\alpha}(I)$ and for any integer number $N\geq 1$, we have

[TABLE]

Proof.

We recall here the expression of the differential operator $\mathcal{L}_{c}^{(\alpha)}$ :

[TABLE]

For any positive integer $r$ and function $f\in H^{r}_{\alpha,0}$ , the derivatives of $\mathcal{L}_{c}^{(\alpha)}.f$ are given by :

[TABLE]

This expression yields directly that :

[TABLE]

Therefore $\mathcal{L}_{c}^{(\alpha)}$ is a continuous operator from $H^{r+2}_{\alpha,0}(I)$ into $H^{r}_{\alpha,0}(I)$.

Then, by induction on $r$, we obtain

[TABLE]

For $f\in H^{r}_{\alpha}(I)$, we write :

[TABLE]

Since $\mathcal{L}_{c}^{(\alpha)}$ is self adjoint on $L^{2}(I,\omega_{\alpha})$,

[TABLE]

Thus

[TABLE]

By writing ${\left\|{f-S_{N}f}\right\|}^{2}_{L^{2}(I,\omega_{\alpha})}=\displaystyle\sum_{n=N+1}^{\infty}|\hat{f_{n}}|^{2}$ one gets (39) Then, we obtain the result for $u$ regular, then for $u\in H^{r+2}_{\alpha}(I)$ by density and continuity. Finally, for $s$ between two even integers, we conclude by interpolation. ∎

In the following, we recall from [13] that for $c>0$ a fixed positive real number, then, for all positive integers $n,\;k$ such that $q=c^{2}/\chi^{\alpha}_{n}(c)<1$ and $k(k+2\alpha+1)+C^{\prime}_{\alpha}c^{2}\leq\chi_{n}^{\alpha}(c)$ , we have

[TABLE]

With $C^{\prime}_{\alpha}$ a constant depends only on $\alpha$, and ${\displaystyle C_{\alpha}=\frac{2^{\alpha}(3/2)^{3/4}(3/2+2\alpha)^{3/4+\alpha}}{e^{2\alpha+3/2}}}$

Lemma 3**.**

Let $c>1$ and $\alpha>0$, then for all positive integers $n,\;k$ such that, $k\leq n/1.9$ and $n\geq m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ we have

[TABLE]

with $\delta$ and $C_{c,\alpha}$ positive constants .

Proof.

By $(\ref{9})$ ,$(\ref{Eq3.1})$ , the inequality $\chi_{n}^{\alpha}(c)\leq n(n+2\alpha+1)+c^{2}$ and the previous remark, one concludes that for $c>0$ and $\alpha>-1$, and for all positive integers $n,\;k$ such that, $k\leq n/A$ and $n\geq Ac$ with $A\geq 1.9$, we have,

[TABLE]

For the appropriate value of $A=1,9$. We have

[TABLE]

Since for all $n\geq m_{\alpha}c$, with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ there are two positive constant $\delta>0$ and $C_{c,\alpha}\geq\frac{C_{\alpha}\left(\frac{ec}{2n-1}\right)^{\frac{\alpha}{2}}}{c^{\frac{({\alpha+2})}{2}}\log(\frac{2n-1}{ec})}$. Then we have (46) ∎

In [11] the author has shown that on $H^{s}_{\alpha,0}$ we can also use the norm

[TABLE]

.

Theorem 5**.**

Let $c>0$ and $\alpha>0$, then there exist generic constants $K>0$ such that, when $N>m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ and $f\in H^{s}_{\alpha}$, we have the inequality

[TABLE]

Proof.

Let $f\in H^{s}_{\alpha,0}$, then

[TABLE]

We have directly that

[TABLE]

Moreover, we can write the function $g$ under the form $g=\displaystyle\sum_{n\in\mathbb{N}}<g,\psi_{n,c}^{(\alpha)}>_{L^{2}(I,\omega_{\alpha})}\psi_{n,c}^{(\alpha)}$

[TABLE]

The last inequality use lemma 3. Then we prove this result for functions $f\in H^{s}_{\alpha,0}$ and we extend it into $H^{s}_{\alpha}$ by density. ∎

Remark 4**.**

The previous proposition is considered as the generalization of a similar result given in [7], in the special case $\alpha=0$

In [7], authors studied the spectral approximation of a function $f\in H^{s}_{per}(I)$. Where $H^{s}_{per}$ is the subspace of $H^{s}_{0}(I)$ that extend into 2-periodic functions of the same regularity. Recall that in this space we define the norm :

[TABLE]

With $\displaystyle b_{k}(f)=\frac{1}{\sqrt{2}}\int_{-1}^{1}f(x)e^{-i\pi kx}dx$ is the coefficient of the Fourier series expansion of $f$. We further prove in the following proposition that the family of GPSWFs are also adapted for the approximation of functions in $f\in H^{s}_{per}$.

Lemma 4**.**

for $c\geq 1$, $\alpha>0$, for all positif real $n,k$ such that $k\leq 0.14n$ and $n\geq m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$. There exist $C_{\alpha,c},\delta>0$ such that

[TABLE]

Proof.

For all $A\geq 1,9$ we have

[TABLE]

For $I_{2}$ since,$|\beta_{m}^{n}|\leq 1$ and by using $(\ref{Eq3.2}),\;(\ref{1})$ and $\eqref{24}$ , one gets

[TABLE]

Where $K$ is a positive constant.

It is clear that for $k\leq\frac{1.2n}{e\pi}\simeq 0.14n$ there is $b>0$ such as

[TABLE]

For $I_{1}$, we have $|\langle e^{ik\pi x},\tilde{P}_{m}^{(\alpha,\alpha)}\rangle_{L^{2}(I,\omega_{\alpha\alpha})}|\leq 1$, so we use (46). For $n>m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ we have

[TABLE]

∎

Proposition 2**.**

Let $c>1$ and $\alpha>0$, then there exist constants $K>0$ and $a>0$ such that, when $N>m_{\alpha}c$ with $m_{\alpha}=4.13\left(1.28+\frac{2\alpha+1}{1.9}\right)^{0.55}$ and $f\in H^{s}_{per}$; $s>0$, we have the inequality

[TABLE]

Where $S_{N}(f)(t)=\displaystyle{\sum_{n<N}}<f,\psi_{n,c}^{\alpha}>_{L^{2}_{\alpha}(I)}\psi_{n,c}^{\alpha}$

Proof.

For $\alpha\geq 0$,we have $H^{s}_{\alpha}(I)\subset H^{s}_{0}(I)$. Let $f\in H^{s}_{per}$, then

[TABLE]

.

[TABLE]

By using the fact that $g\in L^{2}(I,\omega_{\alpha})$ and lemma 3, one gets

[TABLE]

[TABLE]

In the last inequality we use lemma 4 ∎

5 Numerical results

In this section, we give some numerical examples that illustrate the different results of this work.

Example 1:

In this example we illustrate the decay rate of the eigenvalues $(\lambda_{n}^{\alpha}(c))$ given by theorem 3 for different values of $\alpha$. In figure 1 (a), we have plotted the graph of $(\lambda_{n}^{\alpha}(c))$ and in figure 1 (b) we have plotted the graphs of $log(\lambda_{n}^{\alpha}(c))$ versus the graphs of -(2n+1)\log\Big{(}\frac{4n+4\alpha+2}{ec}\Big{)}

Example 2: In this example, for a positive real number $s$, we consider the Brownian motion function

[TABLE]

Here $X_{k}$ is a sequence of independent standard Gaussian random variable. It is well known that $B_{s}\in H^{s^{\prime}-1/2}(I),$ for any $s^{\prime}<s-\frac{1}{2}.$ Here, $H^{s}(I)$ is the Sobolev space over $I,$ with smoothness exponent $s.$ For the special values of $s=1.5$ and $c=5\pi$, and the two values of $N=46$ and $N=90,$ we have computed $S_{N}(B_{s}),$ the truncated series expansion to the order $N,$ of $B_{s}$ in the GPSWFs basis. We have found that

[TABLE]

Moreover, the graphs of $B_{s}$ and of its approximation $S_{90}B_{s}$ are given by the figure 2.

**Example 2:**To illustrate approximation by GPSWFs we have considered the Weierstrass-Mondelbrot function given by :

[TABLE]

For $\lambda>1$ and $s<1$, $M_{s,\lambda}\in H^{s}_{\alpha}(I)$ In this example we have considered the values of $\alpha=0.5$, $c=5\pi$ and $N=95$ then we have computed the projection, over $Span(\psi_{0,c}^{(\alpha)},\cdots,\psi_{N,c}^{(\alpha)})$, $S_{N}(M_{s,\lambda}(x))$. For this purpose, we have computed the different expansion coefficients $\displaystyle{C_{n}(M_{s,\lambda}(x))=\int_{-1}^{1}M_{s,\lambda}(x)\psi_{n,c}^{(\alpha)}(y)\omega_{\alpha}(y)dy}$ which are exactly given thanks to (16). Since $M_{s,\lambda}$ is an odd function , $C_{2n}(M_{s,\lambda})=$ . Moreover

[TABLE]

In figure 3 we have plotted the function $M_{1,2}$ and the graph of the error $M_{1,2}-S_{N}(M_{1,2}(x))$

Moreover in table 1, we give the approximation error ${\left\|{M_{s,2}-S_{N}^{(\alpha)}(M_{s,2})_{L^{2}(I,\omega_{\alpha})}}\right\|}$ of the Weierstrass-Mandelbrot function by GPSWFs for different values of s and $\alpha$.