Weighted Finite Fourier Transform Operator: Uniform Approximations of the Eigenfunctions, Eigenvalues Decay and Behaviour

Abstract

In this paper, we investigate two uniform asymptotic approximations as well as some spectral properties of the eigenfunctions of the weighted finite Fourier transform operator, defined by ${\displaystyle {\mathcal {F}}_c^{(\alpha )} f(x)=\int _{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha }\, dy.}$ Here, $c >0, \alpha \ge -1/2$ are two fixed real numbers. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs) and they are firstly introduced and studied in Wang and Zhang (Appl Comput Harmon Anal 29(3):303–329, 2010). The present study is motivated by the promising concrete applications of the GPSWFs in various scientific area such as numerical analysis, mathematical physics and signal processing. We should mention that these two uniform approximation results of the GPSWFs can be considered as generalizations of the results given in the joint work of one of us (Bonami and Karoui in Constr Approx 43(1):15–45, 2016). As it will be seen, these generalizations require some involved extra work, especially in the case where $\alpha > 1/2.$ By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator ${\mathcal {F}}_c^{(\alpha )}$ in the case where $0<\alpha < 3/2.$ Moreover, by computing the trace and an estimate of the norm of the operator ${\displaystyle {\mathcal {Q}}_c^{(\alpha )}=\frac{c}{2\pi } {\mathcal {F}}_c^{{(\alpha )}^*} \circ {\mathcal {F}}_c^{(\alpha )},}$ we give a lower bound for the counting number of the eigenvalues of $Q_c^{(\alpha )},$ when $c>>1.$ Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Appendices

Appendix 1: Proof of Lemma 3

Let $\alpha > 1/2$ and let $j_{\alpha ,1}, j'_{\alpha ,1}$ and $y_{\alpha ,1}$ denote the first zeros of $J_{\alpha }(x), J'_{\alpha }(x)$ and $Y_{\alpha }(x),$ respectively. It is known, see for example [29, p. 487] that

$$\begin{aligned} \alpha< \sqrt{\alpha (\alpha +2)}< j'_{\alpha ,1}< y_{\alpha ,1}< j_{\alpha ,1}. \end{aligned}$$

Moreover, by using the asymptotic behaviours of $J_{\alpha }(\cdot ), Y_{\alpha }(\cdot ),$ given by (17), one concludes that the function $x J_{\alpha }(x) |Y_{\alpha }(x)|= - x J_{\alpha }(x) Y_{\alpha }(x)$ is positive and bounded over the interval $(0,\alpha ].$

Next, we check that for any $\alpha > 1/2,$ $X_{\alpha } > \alpha ,$ where $X_{\alpha }$ is the first root of $J_{\alpha }(x)+J_{\alpha }(x)=0.$ To this end, we first note that from the Wronskian of $J_{\alpha }, Y_{\alpha },$ given by (18), one concludes that for any $\alpha > -1,$ we have

$$\begin{aligned} \frac{\partial }{\partial x}\left( \frac{-Y_{\alpha }(x)}{J_{\alpha }(x)}\right) = -\frac{2}{\pi x J^2_{\alpha }(x)} <0,\quad x >0. \end{aligned}$$

(79)

Also, from [21, p. 438], $X_{\alpha }$ has the following asymptotic formula, valid for large values of the parameter $\alpha ,$

$$\begin{aligned} X_{\alpha } =\alpha + c (\alpha /2)^{1/3} +O(\alpha ^{-1/3}),\quad c\approx 0.366. \end{aligned}$$

Hence, there exists $\alpha _0 >0 ,$ so that $X_{\alpha } > \alpha ,$ whenever $\alpha \ge \alpha _0.$ Hence, we have

$$\begin{aligned} - \frac{ Y_{\nu }(\nu )}{J_{\nu }(\nu )} \ge - \frac{ Y_{\nu }(X_{\nu })}{J_{\nu }(X_{\nu })}=1,\quad \forall \, \nu \ge \alpha _0, \end{aligned}$$

which means that ${\displaystyle \lim _{\nu \rightarrow +\infty } - \frac{ Y_{\nu }(\nu )}{J_{\nu }(\nu )}\ge 1.}$ On the other hand, from [29, p. 487], we have

$$\begin{aligned} \frac{\partial }{\partial \nu }\left( \frac{-Y_{\nu }(\nu )}{J_{\nu }(\nu )}\right) <0,\quad \nu >0. \end{aligned}$$

Consequently, for any $\alpha \ge 1/2,$ we have

$$\begin{aligned} - \frac{ Y_{\alpha }(\alpha )}{J_{\alpha }(\alpha )} \ge \lim _{\nu \rightarrow +\infty } - \frac{ Y_{\nu }(\nu )}{J_{\nu }(\nu )}\ge 1 = - \frac{ Y_{\alpha }(X_{\alpha })}{J_{\alpha }(X_{\alpha })}, \end{aligned}$$

which means that $X_{\alpha }> \alpha ,$ whenever $\alpha \ge 1/2.$ Hence, for $0<x<\alpha ,$ by integrating (79) over the interval $[x,\alpha ]$ and using the fact that the function $x J^2_{\alpha }(x)$ is increasing, one gets

$$\begin{aligned} -2 x J_{\alpha }(x) Y_{\alpha }(x)= & {} 2 x J^2_{\alpha }(x) \frac{- Y_{\alpha }(\alpha )}{J_{\alpha }(\alpha )}+ \frac{4}{\pi } \int _x^{\alpha } \frac{x J^2_{\alpha }(x)}{t J^2_{\alpha }(t)}\, dt\nonumber \\\le & {} 2\alpha J^2_{\alpha }(\alpha ) \frac{- Y_{\alpha }(\alpha )}{J_{\alpha }(\alpha )}+ \frac{4\alpha }{\pi },\quad 0<x \le \alpha . \end{aligned}$$

(80)

On the other hand, since $2 x |J_{\alpha }(x) Y_{\alpha }(x)\le x \left( J^2_{\alpha }(x)+Y^2_{\alpha }(x)\right) ,$ since this later is decreasing for $\alpha \ge 1/2$ and since $\alpha < X_{\alpha },$ then we have

$$\begin{aligned} \max \left( \sup _{x\in [\alpha , X_{\alpha }]} -x J_{\alpha }(x) Y_{\alpha }(x); \sup _{x\ge X_{\alpha }} x \left( J^2_{\alpha }(x)+Y^2_{\alpha }(x)\right) \right) \le \alpha \left( J^2_{\alpha }(\alpha )+Y^2_{\alpha }(\alpha )\right) . \end{aligned}$$

(81)

Finally, by combining (80) and (81), one gets the desired bound (21). $\square$

Appendix 2: Proof of Lemma 4

The first equality in (32) is a consequence of the following identity, see [22, p. 241]

$$\begin{aligned} \int _0^x t J^2_{\alpha }(t)\, dt = \frac{x^2}{2}\left[ J^2_{\alpha }(x)-J_{\alpha -1}(x)J_{\alpha +1}(x)\right] ,\quad \alpha > -1/2, \end{aligned}$$

combined with the well known identity

$$\begin{aligned} J_{\alpha -1}(x)=\frac{2\alpha }{x}J_{\alpha }(x)-J_{\alpha +1}(x). \end{aligned}$$

Moreover, it has been shown in [16], that for $\alpha \ge -1/2,$ we have

$$\begin{aligned} \sup _{x\ge 0} x^{3/2}\left| J_{\alpha }(x)-\sqrt{\frac{2}{\pi x}} \Big ( \cos \left( x-(\alpha +1/2)\frac{\pi }{2}\right) \Big )\right| \le \frac{4}{5} \mu _{\alpha }. \end{aligned}$$

(82)

Hence, by using the previous inequality, one gets

$$\begin{aligned}&\left| J_{\nu }^2(x)-\frac{2}{\pi x} \cos \Big (x-\Big (\nu +\frac{1}{2}\Big )\frac{\pi }{2}\Big )\right| \le \frac{4}{5 x^{3/2}}\mu _{\alpha } \left( |J_{\nu }(x)|+\sqrt{\frac{2}{\pi x}}\right) ,\nonumber \\&\quad x\ge 1,\quad \nu = \alpha ,\, \alpha +1. \end{aligned}$$

(83)

Moreover, from (82), one gets

$$\begin{aligned} |J_{\nu }(x)| \le \frac{4}{5} \mu _{\nu }+\sqrt{\frac{2}{\pi x}},\quad x\ge 1. \end{aligned}$$

(84)

By using the previous two inequalities, one obtains

$$\begin{aligned} \left| J^2_{\alpha }(x)+J^2_{\alpha +1}(x)-\frac{2}{\pi x}\right| \le \frac{8}{5} \sqrt{\frac{2}{\pi }} \frac{1}{x^2}\Big (\mu _{\alpha }+\mu _{\alpha +1}\Big )+\frac{16}{25 x^3}\Big (\mu ^2_{\alpha }+\mu ^2_{\alpha +1}\Big ),\quad x\ge 1. \end{aligned}$$

Hence, we have

$$\begin{aligned}&{\frac{x^2}{2}\left| J^2_{\alpha }(x)+J^2_{\alpha +1}(x)-\frac{2\alpha }{x} J_{\alpha }(x)J_{\alpha +1}(x)-\frac{2}{\pi x}\right| }\nonumber \\&\quad \le \frac{4}{5} \sqrt{\frac{2}{\pi }} \frac{1}{x^2}\Big (\mu _{\alpha }+\mu _{\alpha +1}\Big )+\frac{8}{25 x}\Big (\mu ^2_{\alpha }+\mu ^2_{\alpha +1}\Big )+|\alpha | x|J_{\alpha }(x)J_{\alpha +1}(x)| ,\quad x \ge 1.\quad \quad \quad \end{aligned}$$

(85)

Finally, by combining (31) and the previous inequality, one gets a bound of $|\eta _{\alpha }(x)|$ for $x\ge 1.$ To get a bound $\eta _{\alpha }(x)$ over the interval [0, 1],  it suffices to note that from (32), we have

$$\begin{aligned} \sup _{x\in [0,1]} |\eta '_{\alpha }(x)|=\sup _{x\in [0,1]} \left| x J^2_{\alpha }(x) -\frac{1}{\pi }\right| \le \max \left( \frac{1}{\pi }, c^2_{\alpha }-\frac{1}{\pi }\right) . \end{aligned}$$

Since $\eta _{\alpha }(0)=0,$ then the previous bound is also valid for ${\displaystyle \sup _{x\in [0,1]} |\eta _{\alpha }(x)| },$ that is

$$\begin{aligned} |\eta _{\alpha }(x)|\le \max \left( \frac{1}{\pi }, c^2_{\alpha }-\frac{1}{\pi }\right) ,\quad 0\le x\le 1. \end{aligned}$$

(86)

Finally, to conclude for the proof of the lemma, it suffices to combine (31), (85) and (86). $\square$