Efficient Methods for the Chebyshev-Type Prolate Spheroidal Wave Functions and Corresponding Eigenvalues

Abstract

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor $e^{\mathrm{i}cxy}$ and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach.

1. Introduction

Prolate Spheroidal Wave Functions of order zero (PSWFs), denoted as ${\left\{{\psi }_n(x;c)\right\}}_{n=0}^{\infty }$ with $c>0$, have been identified as the eigenfunctions of both the truncated Fourier transform (cf. [1])

$${\int }_{-1}^1{e}^{\mathrm{i}cxt}{\psi }_n(t;c)dt={\lambda }_n\left(c\right){\psi }_n(x;c),$$

(1)

and the Sturm–Liouville problem

$$\frac{d}{dx}\left\{(1-x^2)\frac{d}{dx}{\psi }_n(x;c)\right\}+({\chi }_n\left(c\right)-c^2{x}^2){\psi }_n(x;c)=0,$$

(2)

where ${\lambda }_n\left(c\right)$ and ${\chi }_n\left(c\right)$ are the corresponding eigenvalues. This remarkable property endows PSWFs with the character of being band-limited functions with a bandwidth of c [1,2,3,4,5,6,7]. Moreover, for any positive real c, the PSWFs constitute an orthonormal and complete basis in $L^2(-1,1)$, that is

$${\int }_{-1}^1{\psi }_n(x;c){\psi }_m(x;c)dx={\delta }_{mn}:=\left\{\begin{array}{c}0,\mathrm{if}m\ne n,\hfill \\ 1,\mathrm{if}m=n.\hfill \end{array}\right.$$

(3)

Consequently, they are recognized as optimal tools for approximating band-limited functions and have found widespread applications in various scientific domains, such as signal processing and antenna theory (see, for instance, [5,8,9,10,11,12]). The utilization of PSWFs as basis functions in spectral methods is particularly advantageous, especially when addressing problems characterized by wave-like behavior or nearly band-limited solutions.

PSWFs constitute a non-polynomial basis, and therefore no compact form of them can be readily employed in computations. However, they maintain a close connection with the Legendre polynomials ${\left\{P_n\left(x\right)\right\}}_{n=0}^{\infty }$, an orthogonal and complete polynomial system in $L^2[-1,1]$. Specifically, the normalized Legendre polynomials ${\overline{P}}_n\left(x\right)=\sqrt{n+\frac{1}{2}}{P}_n\left(x\right)$ satisfy the Sturm–Liouville problem (2) with $c=0$ and ${\chi }_n\left(0\right)=n(n+1)$, leading to ${\psi }_n(x;0)={\overline{P}}_n\left(x\right)$. For the case $c>0$, the relationship between the PSWFs and normalized Legendre polynomials is established by Slepian [13], Rokhlin and Xiao [14], Osipov et al. [15], and Wang [3]. For a fixed n and a small value of c, it holds that

$${\psi }_n(x;c)={\overline{P}}_n\left(x\right)+\mathcal{O}\left(c^2\right).$$

(4)

Similarly, for any fixed $c>0$ and a large value of n, there exists

$${\psi }_n(x;c)={\overline{P}}_n\left(x\right)+\frac{c^2}{16n}\left({\overline{P}}_{n-2}\left(x\right)-{\overline{P}}_{n+2}\left(x\right)\right)+\mathcal{O}\left(\frac{c^2}{n^2}\right).$$

(5)

Therefore, to evaluate PSWFs and their associated eigenvalues, one can express ${\psi }_n(x;c)$ as a series of the normalized Legendre polynomials:

$${\psi }_n(x;c)=\sum _{k=0}^{\infty }{\gamma }_k^n{\overline{P}}_k\left(x\right),{\gamma }_k^n={\int }_{-1}^1{\psi }_n(x;c){\overline{P}}_k\left(x\right)dx.$$

(6)

The coefficients ${\gamma }_k^n$ decay exponentially due to the analyticity of ${\psi }_n(x;c)$ on the interval $[-1,1]$. Over the past few decades, there has been a growing interest in the development of algorithms for computing PSWFs, associated eigenvalues and their zeros, etc. (see, for instance, [3,15,16,17,18,19,20] and references therein). Additionally, numerous extensions and generalizations of PSWFs have been established to address time-frequency concentration problems on various domains, such as a unit disk [2,21] or sphere [22,23,24].

More recently, for solving the Helmholtz equation in elliptic domains, Wang et al. introduced a series of works on the generalized Prolate Spheroidal Wave Functions (GPSWFs) [25,26]. Let $\alpha >-1$ be a real number, the GPSWFs, denoted by ${\psi }_n^{\left(\alpha \right)}(x;c)$, are defined as the eigenfunction of the singular Sturm–Liouville problem

$${(1-x^2)}^{-\alpha }\frac{d}{dx}\left\{{(1-x^2)}^{\alpha +1}\frac{d}{dx}{\psi }_n^{\left(\alpha \right)}(x;c)\right\}+({\chi }_n^{\left(\alpha \right)}\left(c\right)-c^2{x}^2){\psi }_n^{\left(\alpha \right)}(x;c)=0.$$

(7)

If $\alpha =0$, they reduce to the PSWFs of order zero. Notably, among these generalizations, the GPSWFs of order $\alpha =-1/2$, also referred to as the Chebyshev-type PSWFs (CPSWFs), are of particular importance due to their close relationship with the well-known Chebyshev polynomial system and their wide-ranging applications.

Recalling the Sturm–Liouville equation for the first kind of Chebyshev polynomial (cf. [27], Equation (3.208))

$$\sqrt{1-x^2}\frac{d}{dx}\left\{\sqrt{1-x^2}\frac{d}{dx}{T}_n\left(x\right)\right\}+n^2{T}_n\left(x\right)=0,$$

(8)

we know that ${\psi }_n^{(-1/2)}(x;0)={\overline{T}}_n\left(x\right)$, where ${\overline{T}}_n\left(x\right)=\sqrt{\frac{{\delta }_n}{\pi }}{T}_n\left(x\right)$ denotes the normalized Chebyshev polynomial of the first kind with ${\delta }_0=1$ and ${\delta }_n=2$ for $n\ge 1$. Therefore, the CPSWFs can be viewed as a generalization of the Chebyshev polynomial with a tuning parameter $c>0$. Hereinafter, we rewrite ${\psi }_n^{(-1/2)}(x;c)$ as ${\psi }_n^{(-1/2)}\left(x\right)$ for simplicity. Some basic properties of the CPSWFs are summarized as follows (see also [3,15,25]).

• The CPSWFs are real and smooth on $[-1,1]$. They constitute a complete orthonormal system in $L_w^2[-1,1]$ with $w\left(x\right)={(1-x^2)}^{-1/2}$

  $${\int }_{-1}^1{\psi }_n^{(-1/2)}\left(x\right){\psi }_m^{(-1/2)}\left(x\right)w\left(x\right)dx={\delta }_{nm}.$$

  (9)
  Since the normalized Chebyshev polynomials also constitute an orthonormal basis in $L_w^2[-1,1]$, ${\psi }_n^{(-1/2)}\left(x\right)$ can be expanded by

  $${\psi }_n^{(-1/2)}\left(x\right)=\sum _{k=0}^{\infty }{\beta }_k^n{\overline{T}}_k\left(x\right),{\beta }_k^n={\int }_{-1}^{1}w\left(x\right){\psi }_n^{(-1/2)}\left(x\right){\overline{T}}_k\left(x\right)dx.$$

  (10)
• The eigenvalues of the Sturm–Liouville problem CPSWFs are distinct, real and positive, and can be ordered as

  $$0<{\chi }_0^{(-1/2)}<{\chi }_1^{(-1/2)}<\cdots <{\chi }_n^{(-1/2)}<\cdots .$$

  (11)
  For any $n\ge 0$ and $c>0$, the following uniform bounds for ${\chi }_n^{(-1/2)}$ were established by Wang and Zhang ([25], Lem. 3.1)

  $$n^2<{\chi }_n^{(-1/2)}<n^2+c^2.$$

  (12)
• ${\psi }_n^{(-1/2)}\left(x\right)$ is an even function when n is even, and an odd function when n is odd. There holds the parity

  $${\psi }_n^{(-1/2)}(-x)={(-1)}^n{\psi }_n^{(-1/2)}\left(x\right).$$

  (13)
• ${\psi }_n^{(-1/2)}\left(x\right)$ has exactly n real distinct zeros in the interval $(-1,1)$. When $n\ge 1$, ${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$ has $n-1$ real zeros in $(-1,1)$ that interlace with the n zeros of ${\psi }_n^{(-1/2)}\left(x\right)$. We define the roots of $(1-x^2){\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$ as the Chebyshev–Prolate–Lobatto points (CPL points). In computation, we use Newton’s iteration method with zeros of Chebyshev polynomial as the initial points [16].

This work is devoted to the computation of eigenvalues of CPSWF’s integral operator (14) and demonstrates that this rule outperforms the formula given by (17) when c is small. Furthermore, FFT and barycentric interpolation can be employed to evaluate the CPSWF values efficiently, which play an important role in different numerical applications based on these later. The structure of the paper is organized as follows. In Section 2, we derive efficient methods for computing the eigenvalues of an integral operator. Section 3 investigates the fast interpolation method for the evaluation of CPSWFs and their related quantities, and its efficiency is illustrated by some numerical examples in Section 4. Section 5 is devoted to some applications in highly oscillatory BVP and eigenvalue problems. The final section contains some concluding remarks.

2. The Eigenvalues of the Integral Operator

Aligned with the Sturm–Liouville problem (7), Wang and Zhang [25] demonstrated that ${\psi }_n^{(-1/2)}\left(x\right)$ satisfies the integral equation

$${\mathrm{i}}^n{\lambda }_n{\psi }_n^{(-1/2)}\left(x\right)={\int }_{-1}^1{e}^{\mathrm{i}cxt}{\psi }_n^{(-1/2)}\left(t\right)w\left(t\right)dt:={\mathcal{F}}_c\left({\psi }_n^{(-1/2)}\right)\left(x\right).$$

(14)

Alternatively, this can be expressed as

$${\mu }_n{\psi }_n^{(-1/2)}\left(x\right)={\int }_{-1}^1{J}_0\left(c\right|t-x\left|\right){\psi }_n^{(-1/2)}\left(t\right)w\left(t\right)dt:={\mathcal{Q}}_c\left({\psi }_n^{(-1/2)}\right)\left(x\right).$$

(15)

where $J_{\nu }\left(x\right)$ denotes the Bessel function of the first kind with order $\nu$. It is evident that ${\lambda }_n$ (modulo the factor ${\mathrm{i}}^n$) and ${\mu }_n$ represent, respectively, the eigenvalues corresponding to ${\psi }_n^{(-1/2)}\left(x\right)$ for the integral operators ${\mathcal{F}}_c$ and ${\mathcal{Q}}_c$, and we have ${\mu }_n={\lambda }_n^2/\pi$.

As demonstrated in [25], all ${\lambda }_n$ are positive and arranged in a decreasing series ${\lambda }_0>{\lambda }_1>\cdots >{\lambda }_n>\cdots >0$. For fixed $c>0$, ${\lambda }_n$ decays super-geometrically with respect to n, which shows the asymptotic behavior that ([25], (3.61))

$${\lambda }_n\simeq \sqrt{\frac{2\pi }{n+1}}{\left(\frac{e·c}{4n}\right)}^n,n\gg 1.$$

(16)

Using the parity of PSWFs and setting $x=0$, the numerical evaluations of ${\lambda }_n$ are given by ([25], (3.41))

$${\lambda }_n=\left\{\begin{array}{ccc}\frac{\sqrt{\pi }{\beta }_0^n}{{\mathrm{i}}^n{\psi }_n^{(-1/2)}\left(0\right)},\hfill & \hfill & \mathrm{if}n\mathrm{is}\mathrm{even},\hfill \\ \frac{c\sqrt{\pi }{\beta }_1^n}{{\mathrm{i}}^{n-1}\sqrt{2}{\partial }_x{\psi }_n^{(-1/2)}\left(0\right)},\hfill & \hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(17)

This formula provides a scheme for computing ${\lambda }_n$ after numerically solving for ${\beta }_0^n$, ${\beta }_1^n$, ${\psi }_n^{(-1/2)}\left(0\right)$, and ${\partial }_x{\psi }_n^{(-1/2)}\left(0\right)$. We know that by a suitable truncation of (10), a Bouwkamp-type algorithm can be used for the numerical computation of ${\psi }_n^{(-1/2)}\left(0\right)$ and ${\partial }_x{\psi }_n^{(-1/2)}\left(0\right)$, and then ${\beta }_0^n$ and ${\beta }_1^n$ are derived by solving an eigen-system of a symmetric five-diagonal matrix. For more details of the algorithm, please refer to [25,28]. However, this process may lead to round-off errors, especially for small values of c. Numerical evidence is demonstrated in Figure 1, where unexpected instability is observed when $n=90$ and $c=1$, $c=45$.

Now, we introduce a distinct method for accurate computation of the eigenvalues ${\lambda }_n$ when c is small, denoted as the expansion method. The basic idea of this approach is the utilization of the expansion formula [29]

$$e^{\mathrm{i}cxy}=2\sum _{n=0}^{\infty }{ }^{\prime }{\mathrm{i}}^n{J}_n\left(cx\right)T_n\left(y\right)=2\sum _{n=0}^{\infty }{ }^{\prime }\sqrt{\frac{\pi }{{\delta }_n}}{\mathrm{i}}^n{J}_n\left(cx\right){\overline{T}}_n\left(y\right),$$

(18)

where the prime denotes a sum whose first term is halved, and ${\delta }_0=1$ and ${\delta }_n=2$ for $n\ge 1$. By exploiting the properties of Chebyshev polynomials and Bessel functions, it is sufficient to use a submatrix with a low order N to obtain an accurate approximation of N eigenvalues of ${\mathcal{F}}_c\left({\psi }_n^{-1/2}\right)$.

We first provide some uniform bounds on ${\psi }_n^{(-1/2)}\left(x\right)$ in the following Lemmas.

Lemma 1. 

Suppose $n\ge 0$ is an integer, and x and y are two arbitrary extreme points of ${\psi }_n^{(-1/2)}\left(x\right)$ (the zeros of ${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$) in $(-1,1)$. If $\left|x\right|<\left|y\right|$, then

$$|{\psi }_n^{(-1/2)}\left(x\right)|<|{\psi }_n^{(-1/2)}\left(y\right)|.$$

(19)

Proof. 

Introduce $p\left(t\right)=1-t^2$, $q\left(t\right)={\gamma }_n-c^2{t}^2$, and define

$$U\left(t\right)={\left({\psi }_n^{(-1/2)}\left(t\right)\right)}^2+\frac{p\left(t\right)}{q\left(t\right)}{\left({\partial }_t{\psi }_n^{(-1/2)}\left(t\right)\right)}^2.$$

(20)

Then, by differentiating $U\left(t\right)$ with respect to t and combining with (8), we obtain

$$U^{\prime }\left(t\right)=\frac{2c^{2}t(1-t^2)}{{({\gamma }_n-c^2{t}^2)}^2}{\left({\psi }_n^{{(-1/2)}^{\prime }}\left(t\right)\right)}^2.$$

(21)

which implies that $U\left(t\right)$ is increasing when $t\in (0,1)$. Thus, for extreme points $x,y\in (0,1)$ and $x<y$, we have

$$|{\psi }_n^{(-1/2)}\left(x\right)|<|{\psi }_n^{(-1/2)}\left(y\right)|.$$

(22)

By noting that the $|{\psi }_n^{(-1/2)}\left(x\right)|$ is even in $(-1,1)$, the Lemma is proved. □

Corollary 1. 

Suppose that $c>0$ is a real number, and $n\ge 0$ is an integer. Then,

$$\parallel {\psi }_n^{(-1/2)}{\parallel }_{L^{\infty }[-1,1]}=\left|{\psi }_n^{(-1/2)}\left(1\right)\right|.$$

(23)

Our task now is to show that the eigenvalues of (14) coincide with the eigenvalues of an infinite matrix T.

Theorem 1. 

For any integers $k,j\ge 0$, let $T={\left[t_{jk}\left(c\right)\right]}_{j,k\ge 0}$ be the infinite matrix with components

$$t_{kj}\left(c\right)=2\pi r_j{\mathrm{i}}^j\sqrt{\frac{{\delta }_k}{{\delta }_j}}{J}_{(j+k)/2}\left(\frac{c}{2}\right)J_{|j-k|/2}\left(\frac{c}{2}\right),$$

(24)

where $r_0=1/2$, ${\delta }_0=1$, $r_j=1$ and ${\delta }_j=2$, for $j\ge 1$. Then, the eigenvalues ${\mathrm{i}}^n{\lambda }_n$ of the integral operator (14) coincide with the eigenvalues of T.

Proof. 

Due to Corollary 1, we know that for any integer $n\ge 0$, ${\psi }_n^{(-1/2)}\left(x\right)$ satisfies

$$\parallel {\psi }_n^{(-1/2)}{\parallel }_{L^{\infty }[-1,1]}=\left|{\psi }_n^{(-1/2)}\left(1\right)\right|.$$

(25)

It follows that the series $2{\sum }_{j=0}^{\infty }{r}_j\sqrt{\frac{\pi }{{\delta }_j}}{\mathrm{i}}^j{J}_j\left(cx\right){\overline{T}}_j\left(y\right){\psi }_n^{(-1/2)}\left(y\right)$ converges uniformly to $e^{\mathrm{i}cxy}{\psi }_n^{(-1/2)}\left(y\right)$ over $[-1,1]$. Consequently, by substituting the expansion (18) into (14), we have

$$\begin{array}{cc}\hfill {\mathrm{i}}^n{\lambda }_n{\psi }_n^{(-1/2)}\left(x\right)=& {\int }_{-1}^1{e}^{\mathrm{i}cxy}{\psi }_n^{(-1/2)}\left(y\right)w\left(y\right)dy\hfill \\ \hfill =& 2\sum _{j=0}^{\infty }{r}_j\sqrt{\frac{\pi }{{\delta }_j}}{\mathrm{i}}^j{J}_j\left(cx\right){\int }_{-1}^1{\overline{T}}_j\left(y\right){\psi }_n^{(-1/2)}\left(y\right)w\left(y\right)dy.\hfill \end{array}$$

(26)

Recalling the Chebyshev expansion of ${\psi }_n^{(-1/2)}\left(x\right)$ in (10), (26) can be rewritten as follows

$$\begin{array}{cc}\hfill {\mathrm{i}}^n{\lambda }_n{\psi }_n^{(-1/2)}\left(x\right)=& 2\sum _{k=0}^{\infty }{\beta }_k^n\sum _{j=0}^{\infty }{r}_j\sqrt{\frac{\pi }{{\delta }_j}}{\mathrm{i}}^j{J}_j\left(cx\right){\int }_{-1}^1{\overline{T}}_j\left(y\right){\overline{T}}_k\left(y\right)w\left(y\right)dy.\hfill \end{array}$$

(27)

Multiplying both sides by $w\left(x\right)$ and ${\overline{T}}_{k^{\prime }}\left(x\right)$, $k^{\prime }=0,1,...$, then integrating over $[-1,1]$, yields

$$\begin{array}{cc}\hfill & {\mathrm{i}}^n{\lambda }_n\sum _{k=0}^{\infty }{\beta }_k^n{\int }_{-1}^1{\overline{T}}_k\left(x\right){\overline{T}}_{k^{\prime }}\left(x\right)w\left(x\right)dx\hfill \\ \hfill & =2\sum _{k=0}^{\infty }{\beta }_k^n\sum _{j=0}^{\infty }{r}_j\sqrt{\frac{\pi }{{\delta }_j}}{\mathrm{i}}^j{\int }_{-1}^1{J}_j\left(cx\right){\overline{T}}_{k^{\prime }}\left(x\right)w\left(x\right)dx{\int }_{-1}^1{\overline{T}}_j\left(y\right){\overline{T}}_k\left(y\right)w\left(y\right)dy.\hfill \end{array}$$

(28)

It is clear that ${\lambda }_n$ are the eigenvalues of ${\mathcal{F}}_c^{-1/2}\left({\psi }_n\right)$ if and only if they are eigenvalues of the infinite matrix T with coefficients

$$t_{kj}\left(c\right)=2r_j{\mathrm{i}}^j\sqrt{\frac{\pi }{{\delta }_j}}{\int }_{-1}^1{J}_j\left(cx\right){\overline{T}}_k\left(x\right)w\left(x\right)dx=2r_j{\mathrm{i}}^j\sqrt{\frac{{\delta }_k}{{\delta }_j}}{\int }_{-1}^1{J}_j\left(cx\right)T_k\left(x\right)w\left(x\right)dx.$$

Applying the formula given by ([30], p. 803)

$${\int }_0^1{(1-x^2)}^{-1/2}{T}_n\left(x\right)J_v\left(xy\right)dx=\frac{\pi }{2}{J}_{\frac{1}{2}(v+n)}\left(\frac{y}{2}\right)J_{\frac{1}{2}|v-n|}\left(\frac{y}{2}\right),$$

combining with

$$J_n(-x)={(-1)}^n{J}_n\left(x\right),T_n(-x)={(-1)}^n{T}_n\left(x\right),$$

we see that the coefficients of matrix T are simply given by

$$t_{kj}\left(c\right)=2r_j{\mathrm{i}}^j\sqrt{\frac{{\delta }_k}{{\delta }_j}}{\int }_{-1}^1{J}_j\left(cx\right)T_k\left(x\right)w\left(x\right)dx=2\pi r_j{\mathrm{i}}^j\sqrt{\frac{{\delta }_k}{{\delta }_j}}{J}_{\frac{1}{2}(j+k)}\left(\frac{c}{2}\right)J_{\frac{1}{2}|j-k|}\left(\frac{c}{2}\right).$$

(29)

This completes the proof. □

The following lemma provides us with the decay of the components $t_{kj}\left(c\right)$.

Theorem 2. 

For any $c>0$ and positive integers k, j, we have

$$t_{kj}\left(c\right)\le \left\{\begin{array}{ccc}\frac{2\sqrt{2}}{\sqrt{(j-k)(j+k)}}{\left(\frac{ec}{2(j+k)}\right)}^{\frac{j+k}{2}}{\left(\frac{ec}{2|j-k|}\right)}^{\frac{|j-k|}{2}},\hfill & \hfill & ifj\ne k,\hfill \\ \sqrt{\frac{2\pi }{j}}{\left(\frac{ec}{4j}\right)}^j,\hfill & \hfill & ifj=k.\hfill \end{array}\right.$$

(30)

Proof. 

For a fixed x and a large j, there exists the asymptotic formula ([31], p. 365)

$$J_j\left(x\right)\sim \frac{1}{\sqrt{2\pi j}}{\left(\frac{ex}{2j}\right)}^j.$$

Consequently, if $\left|\frac{ex}{2j}\right|\le 1$, the values of $J_j\left(x\right)$ decrease very rapidly with increasing j.

In fact, according to (29), for $j\ne k$,

$$t_{kj}\left(c\right)\le \frac{2\sqrt{2}}{\sqrt{(j-k)(j+k)}}{\left(\frac{ec}{2(j+k)}\right)}^{\frac{j+k}{2}}{\left(\frac{ec}{2|j-k|}\right)}^{\frac{|j-k|}{2}}.$$

Using the fact that $J_0\left(\frac{c}{2}\right)\le 1$, we have

$$t_{jj}\left(c\right)\le \sqrt{\frac{2\pi }{j}}{\left(\frac{ec}{4j}\right)}^j.$$

Thus, we obtain the results (30). □

The coefficients of the discretization matrix T exhibit rapid decay towards zero. Consequently, utilizing a submatrix $T_N$ with a low order N proves to be adequate for obtaining accurate approximations of N eigenvalues.

In Figure 2, we illustrate several eigenvalues of the integral operator obtained using the formula (17), the expansion method (Theorem 1), and asymptotic behavior (16) for small values of c when $n=200$. It is evident that the expansion method yields satisfactory numerical approximations to the eigenvalues and demonstrates superior stability and accuracy compared to the Formula (17).

3. Evaluation of CPSWF and Related Quantities

To achieve efficient computation of CPSWFs, we employ two primary strategies: applying the fast Fourier transform (FFT) to evaluate CPSWF values at Chebyshev points, and utilizing barycentric interpolation to accurately approximate CPSWF values or derivatives across the interval $[-1,1]$.

Initially, we express ${\psi }_n^{(-1/2)}\left(x\right)$ using normalized Chebyshev polynomials

$${\psi }_n^{(-1/2)}\left(x\right)=\sum _{k=0}^{\infty }{\beta }_k^n{\overline{T}}_k\left(x\right),{\beta }_k^n={\int }_{-1}^{1}w\left(x\right){\psi }_n^{(-1/2)}\left(x\right){\overline{T}}_k\left(x\right)dx.$$

(31)

This representation enables us to exploit the advantageous properties of Chebyshev polynomials for efficiently computing function values, derivatives, and integrals of CPSWFs. Substituting (31) into (8) and utilizing the properties of Chebyshev polynomials, such as

$$(1-x^2){\overline{T}}_n^{″}\left(x\right)-x{\overline{T}}_n^{\prime }\left(x\right)+n^2{\overline{T}}_n\left(x\right)=0,$$

(32)

and the recurrence relation

$${\overline{T}}_{n+1}\left(x\right)=2x{\overline{T}}_n\left(x\right)-{\overline{T}}_{n-1}\left(x\right),n\ge 1,$$

(33)

we derive an equivalent eigen-problem [25]

$$(\mathbf{B}-{\gamma }_n·\mathbf{I}){\overrightarrow{\beta }}^n=0,$$

(34)

where ${\overrightarrow{\beta }}^n=({\beta }_0^n,{\beta }_1^n,{\beta }_2^n,\dots )$ and $\mathbf{B}$ is a symmetric five-diagonal matrix with three nonzero diagonals given by

$$B_{k,k}=k^2+\frac{c^2}{2},B_{k,k+2}=B_{k+2,k}=\frac{c^2}{4}.$$

(35)

For computation, truncating the infinite series in (31) is necessary, and a recommended truncation term is $p=2n+30$ as suggested in [25].

Once we have obtained the coefficients $\left\{{\beta }_k^n\right\}$, the values of ${\psi }_n^{(-1/2)}\left(x\right)$ at arbitrary points ${\left\{x_j\right\}}_{j=0}^N$ can be obtained using the Direct method, which is expressed as

$$\psi =\overline{T}·\beta ,$$

(36)

where $\psi ={({\psi }_n^{(-1/2)}\left(x_0\right),{\psi }_n^{(-1/2)}\left(x_1\right),\dots ,{\psi }_n^{(-1/2)}\left(x_N\right))}^T$, $\beta ={({\beta }_0^n,{\beta }_1^n,\dots ,{\beta }_p^n)}^T$, ${\overline{T}}_{ij}={\overline{T}}_j\left(x_i\right)$. The values of $\left\{{\overline{T}}_j\left(x_i\right)\right\}$ are evaluated using the recurrence relation (33).

Additionally, it is noteworthy that ${\psi }_n^{(-1/2)}\left(x\right)$ can be expressed as an expansion

$${\psi }_n^{(-1/2)}\left(x\right)=\sqrt{\frac{1}{\pi }}{\beta }_0^n+\sqrt{\frac{2}{\pi }}\sum _{k=1}^p{\beta }_k^n{T}_k\left(x\right),$$

(37)

where p is the degree of the expansion. According to [32], the integration of ${\psi }_n^{(-1/2)}\left(x\right)$ has a series expansion of the form

$${\int }_{-1}^x{\psi }_n^{(-1/2)}\left(t\right)dt\approx \sum _{k=0}^{p+1}{b}_k{T}_k\left(x\right),$$

(38)

with the coefficients

$$\begin{array}{cc}\hfill & b_0=\sum _{j=1}^{p+1}{(-1)}^{j+1}·b_j,b_1=\frac{1}{2\sqrt{\pi }}({\beta }_0^n-\sqrt{2}{\beta }_2^n),\hfill \\ \hfill & b_k=\frac{1}{\sqrt{2\pi }k}·({\beta }_{k-1}^n-{\beta }_{k+1}^n),2\le k\le p+1.\hfill \end{array}$$

(39)

Similarly, the derivative of ${\psi }_n^{(-1/2)}\left(x\right)$ can be approximated by

$${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)\approx \sum _{k=0}^{p-1}{d}_k{T}_k\left(x\right),$$

(40)

with the coefficients

$$d_k=\sqrt{\frac{2}{\pi }}\sum _{\begin{array}{c}j=k+1\\ k-jisodd\end{array}}^{p}2j{\beta }_j^n,0\le k\le p-1.$$

(41)

The second-order derivative of ${\psi }_n^{(-1/2)}\left(x\right)$ can be approximated by

$${\partial }_{xx}{\psi }_n^{(-1/2)}\left(x\right)\approx \sum _{k=0}^{p-2}{c}_k{T}_k\left(x\right),$$

(42)

with the coefficients

$$\begin{array}{c}\hfill c_k=\sum _{\begin{array}{c}i=k+2\\ i+kiseven\end{array}}i(i^2-k^2)\sqrt{\frac{2}{\pi }}{\beta }_i^n.\end{array}$$

(43)

Based on the provided formulas, it is evident that by combining with the fast Fourier transform (FFT) [33], we can compute the values of ${\psi }_n^{(-1/2)}\left(x\right)$, the integral ${\int }_{-1}^x{\psi }_n^{(-1/2)}\left(t\right)dt$, and the first- or second-order derivatives of ${\psi }_n^{(-1/2)}\left(x\right)$ at the Clenshaw–Curtis points $\{x_j=cos(j\pi /N),j=0,1,\dots N\}$ with $\mathcal{O}(NlogN)$ operations. Then, by integrating barycentric interpolation, we propose a Fast interpolation method to compute the values of ${\psi }_n^{(-1/2)}\left(x\right)$ at arbitrary points. Since ${\psi }_n^{(-1/2)}\left(x\right)$ is analytic [25], barycentric interpolation [34,35,36,37] can be used to approximate ${\psi }_n^{(-1/2)}\left(x\right)$ by

$$\left(\mathcal{I}{\psi }_n^{(-1/2)}\right)\left(x\right)=\sum _{j=0}^N\frac{w_j}{x-x_j}{\psi }_n^{(-1/2)}\left(x_j\right)/\sum _{k=0}^N\frac{w_k}{x-x_k},$$

(44)

where ${\left\{x_j\right\}}_{j=0}^N$ is the Clenshaw–Curtis point set and the corresponding barycentric weights are

$$w_j={(-1)}^j{\widehat{w}}_j,{\widehat{w}}_j=\left\{\begin{array}{ccc}1/2,\hfill & \hfill & \mathrm{if}j=0\mathrm{or}N,\hfill \\ 1,\hfill & \hfill & \mathrm{if}j=1,\dots ,N-1.\hfill \end{array}\right.$$

(45)

Similarly, the barycentric interpolation formula can be extended to compute derivatives at arbitrary points, yielding

$$\left(\mathcal{I}{\partial }_x{\psi }_n^{(-1/2)}\right)\left(x\right)=\sum _{j=0}^N\frac{w_j}{x-x_j}{\partial }_x{\psi }_n^{(-1/2)}\left(x_j\right)/\sum _{k=0}^N\frac{w_k}{x-x_k}.$$

(46)

In Table 1, Table 2 and Table 3, we present a comparison of the time required to evaluate the values of ${\psi }_n^{(-1/2)}\left(x\right)$ and ${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$ at some given point sets on $[-1,1]$ using the Fast Interpolation method and the Direct method (36). The results indicate the efficiency of our proposed method.

Remark 1. 

As demonstrated in [38] for the PSWF case, we introduced Barycentric prolate interpolation and differentiation, which is essential in various applications of spectral methods. One crucial aspect is the computation of prolate barycentric weights. When the interpolation points $x_j$ are chosen as the roots of $(1-x^2){\psi }_n^{(-1/2)}\left(x\right)$, we define the prolate barycentric weights straightforwardly as

$$w_j=\frac{1}{-2x_j{\psi }_n^{(-1/2)}\left(x_j\right)+(1-x_j^2){\partial }_x{\psi }_n^{(-1/2)}\left(x_j\right)}.$$

(47)

Similarly, when we choose the CPL points $\left\{x_j\right\}$ (zeros of $(1-x^2){\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$) as the interpolation points, their barycentric weights are given by

$$w_j=\frac{1}{-x_j{\partial }_x{\psi }_n^{(-1/2)}\left(x_j\right)+(c^2{x}_j^2-{\gamma }_j){\psi }_n^{(-1/2)}\left(x_j\right)}.$$

(48)

From a numerical standpoint, computing prolate barycentric weights only requires the function values ${\psi }_n^{(-1/2)}\left(x_j\right)$ and its derivatives ${\partial }_x{\psi }_n^{(-1/2)}\left(x_j\right)$. It is worth noting that the fast interpolation method (46) can be employed to enhance its efficiency.

Remark 2. 

Indeed, it is important to recall that the fast multipole method (FMM) [39,40] is a highly efficient algorithm for evaluating series like (46). This provides us with an additional advantage: when combined with FMM, the fast interpolation method becomes significantly more economical compared to the $\mathcal{O}\left(N^2\right)$ cost of direct summation (36).

4. Numerical Results

In this section, we illustrate the efficiency of the proposed methods. All numerical results in this paper are obtained using Matlab R2014a on a desktop computer equipped with 4.0 GB of RAM and two Core2 processors (64 bit) running at 3.17 GHz, operating on the Windows 7 operating system.

Example 1. 

Table 1 and Table 2 compare the efficiency of evaluating the values and derivatives of CPSWFs at Clenshaw–Curtis points using two different methods ($c=N/2$). The fast interpolation method utilizes the FFT, while the Direct method is based on the recurrence relation (33) and direct calculation (36). Assuming that the coefficients $\left\{{\beta }_k^n\right\}$ are available, we consider the CPU time for the evaluation step. The Absolute errors denote the maximum absolute difference between the results obtained by the two methods for different n. It is observed that the fast interpolation method outperforms the Direct calculation.

Table 1. Time(s) of evaluating the CPSWF values at CC points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
200	0.002	0.015	$2.88\times {10}^{-13}$
400	0.004	0.036	$8.18\times {10}^{-13}$
800	0.008	0.181	$2.30\times {10}^{-12}$
1600	0.030	1.310	$3.91\times {10}^{-12}$

Table 1. Time(s) of evaluating the CPSWF values at CC points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
200	0.002	0.015	$2.88\times {10}^{-13}$
400	0.004	0.036	$8.18\times {10}^{-13}$
800	0.008	0.181	$2.30\times {10}^{-12}$
1600	0.030	1.310	$3.91\times {10}^{-12}$

Table 2. Time(s) of evaluating the ${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$ at CC points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
200	0.02	0.02	$1.47\times {10}^{-8}$
400	0.03	0.06	$8.12\times {10}^{-8}$
800	0.11	0.20	$2.88\times {10}^{-6}$
1600	0.44	1.26	$3.49\times {10}^{-5}$

Table 2. Time(s) of evaluating the ${\partial }_x{\psi }_n^{(-1/2)}\left(x\right)$ at CC points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
200	0.02	0.02	$1.47\times {10}^{-8}$
400	0.03	0.06	$8.12\times {10}^{-8}$
800	0.11	0.20	$2.88\times {10}^{-6}$
1600	0.44	1.26	$3.49\times {10}^{-5}$

Example 2. 

We evaluate the CPSWF values at random points on $[-1,1]$ by two different methods in Table 3. The Fast interpolation method is using FFT and barycentric interpolation (46). The direct method is still computed by the recurrence relation (33) and direct calculation (36). As with the above example, we compare the CPU time and the Absolute errors . We observe the same behavior as in the previous cases.

Table 3. Time(s) of evaluating the CPSWF values at random points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
160	0.007	0.008	$9.45\times {10}^{-14}$
320	0.017	0.021	$2.30\times {10}^{-13}$
640	0.043	0.110	$4.64\times {10}^{-13}$
1280	0.159	0.740	$1.05\times {10}^{-12}$
2560	0.651	4.913	$4.14\times {10}^{-12}$
5120	2.719	53.400	$7.97\times {10}^{-12}$

Table 3. Time(s) of evaluating the CPSWF values at random points.

n	CPU Time (s)		Absolute Errors
	Fast Interpolation Method	Direct Method
160	0.007	0.008	$9.45\times {10}^{-14}$
320	0.017	0.021	$2.30\times {10}^{-13}$
640	0.043	0.110	$4.64\times {10}^{-13}$
1280	0.159	0.740	$1.05\times {10}^{-12}$
2560	0.651	4.913	$4.14\times {10}^{-12}$
5120	2.719	53.400	$7.97\times {10}^{-12}$

5. Applications

Building upon the efficient CPSWF calculations outlined in the previous section, this section explores further applications. From numerical perspectives, we emphasize that employing the spectral collocation scheme with CPSWFs as basis functions surpasses the Chebyshev polynomial-based method in handling oscillatory boundary value problems (BVPs) and model eigen-problems.

Example 3. 

Consider the second-order boundary value problem (BVP) given by

$$u^{″}\left(x\right)+5u^{\prime }\left(x\right)+10000u\left(x\right)=-500cos\left(100x\right)e^{-5x},x\in [0,1],$$

(49)

with boundary conditions

$$u\left(0\right)=0,u\left(1\right)=sin\left(100\right)e^{-5}.$$

(50)

The exact solution is

$$u\left(x\right)=sin\left(100x\right)e^{-5x}.$$

(51)

Traditional collocation methods yield the system [41]

$$({\mathbf{D}}_{in}^{\left(2\right)}+5{\mathbf{D}}_{in}^{\left(1\right)}+10000I)\mathbf{u}=\mathbf{f}-{\mathbf{u}}_b,$$

where ${\mathit{D}}^{\left(1\right)}$ and ${\mathit{D}}^{\left(2\right)}$ denote the first- and second-order differential matrices, $\mathbf{f}={\left(f(x_1,...,f\left(x_{n-1}\right))\right)}^T$, $\mathbf{u}$ is a vector of unknowns, and ${\mathbf{u}}_b\left(k\right)=u_{-}(d_{k0}^{\left(2\right)}+5d_{k0}^{\left(1\right)})+u_{+}(d_{kn}^{\left(2\right)}+5d_{kn}^{\left(1\right)})$. The differential matrix, computed as described in [38], is defined by

$$\begin{array}{c}\hfill {\mathit{D}}_{kj}^{\left(1\right)}=\frac{w_j/w_k}{x_k-x_j},k\ne j,{\mathit{D}}_{kk}^{\left(1\right)}=-\sum _{j=0,j\ne k}^N\frac{w_j/w_k}{x_k-x_j},\end{array}$$

(52)

$$\begin{array}{cc}\hfill & {\mathit{D}}_{kj}^{\left(2\right)}=-2\frac{w_j/w_k}{x_k-x_j}\left(\sum _{l\ne k}\frac{w_l/w_k}{x_k-x_l}+\frac{1}{x_k-x_j}\right),k\ne j,\hfill \\ \hfill & {\mathit{D}}_{kk}^{\left(2\right)}=-\sum _{j=0,j\ne k}^N\left(-2\frac{w_j/w_k}{x_k-x_j}(\sum _{l\ne k}\frac{w_l/w_k}{x_k-x_l}+\frac{1}{x_k-x_j})\right),\hfill \end{array}$$

(53)

where ${\left\{x_j\right\}}_{j=0}^N$ and ${\left\{w_j\right\}}_{j=0}^N$ represent the collocation points and their corresponding barycentric weights, respectively. When ${\left\{x_j\right\}}_{j=0}^n$ are CPL points, the weights are given in (48). The Clenshaw–Curtis points $\{x_j=cos(j\pi /N),j=0,1,\dots N\}$ and their barycentric weights can be obtained by (45).

Solving the systems using MATLAB’s GMRES solver, we compare the prolate-based method with the Chebyshev polynomial-based method based on their convergence behavior. As observed in Figure 3, the prolate-based spectral collocation method exhibits good performance, with $c=n$ yielding superior accuracy.

Example 4. 

Let us consider the Laplace eigenvalue problem in two dimensions defined as

$${-▵u=\mu u\mathrm{\Omega }=[-1,1]\times [-1,1],u|}_{\mathrm{\Omega }}=0,$$

(54)

and the eigenvalues of this problem are

$${\mu }_{kj}=\frac{(k^2+j^2){\pi }^2}{4},u_{kj}(x,y)=sin\frac{k\pi (x+1)}{2}sin\frac{j\pi (y+1)}{2},k,j\ge 1.$$

To numerically solve this problem using a spectral method [42], we construct a tensor product grid based on prolate points. Utilizing the Kronecker product, the discrete Laplacian becomes

$$L_N=-I\otimes D_{in}^{\left(2\right)}-D_{in}^{\left(2\right)}\otimes I,$$

(55)

and the corresponding eigenvalue problem is

$$L_N\widehat{u}=\widehat{\mu }\widehat{u},$$

(56)

where I denotes the identity matrix.

In our computation, we set $c=N$, a choice recommended by [10]. Notably, when c approximates N, the function ${\psi }_N^{(-1/2)}\left(x\right)$ exhibits nearly uniform oscillatory behavior, and the prolate points are quasi-uniformly distributed. In Figure 4, we compare the numerical outcomes obtained by Chebyshev differentiation and prolate differentiation (53) with $N=80$. We assess the relative errors using

$$\widetilde{e_k}:=\frac{|\widehat{{\mu }_k}-{\mu }_k|}{|{\mu }_k|}.$$

A numerical comparison shows that for the Laplace eigenvalue problem in two dimensions, approximately ${\left(\frac{2}{\pi }\right)}^2$ eigenvalues are reliably computed using the Chebyshev formulation [43] compared with about $54\%$ for the prolate differentiation.

6. Conclusions

In this paper, based on the powerful properties at Chebyshev polynomials, we propose the efficient mean for evaluating the eigenvalues associated with the integral operator of CPSWFs. Moreover, together with FFT and the barycentric interpolation formula, we provide the fast implementation of values and derivatives of CPSWFs, which can be used in the spectral method for the BVP and eigenvalue problem. We also give some numerical examples that illustrate the results of this work.