2
$\begingroup$

The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation: $$\int_{-T}^T\varphi_n(x) \text{sinc}(t-x) dx = \lambda_n \varphi_n(t)$$

where $\text{sinc}(t) = \sin(\pi t)/ \pi t$. Alternatively (as discovered by Slepian et al.) they are also the eigenfunctions of the following differential operator: $$(1-t^2)\frac{d^2\varphi_n}{dt^2}-2t\frac{d\varphi_n}{dt} -(2 \pi T \Omega)^2t^2 \varphi_n = \mu_n \varphi_n$$

What I would like to prove is that the following $p \times p$ matrix-valued function: $$ \mathbf{S}_{i, k}(u) = \sum_{r = 0}^{R-1}\varphi_r\left(\frac{T}{\Omega}\left(u+\frac{i}{p}\right)\right)\varphi_r\left(\frac{T}{\Omega}\left(u+\frac{k}{p}\right)\right) i,k = 0,\ldots,p-1$$ defined for $u \in [0, 1/p)$ has a positive determinant $a.e.$ on $[0,1/p)$ i.e.,

$$ \text{det}(\mathbf{S}(u)) = K > 0 \text{ } a.e. \text{ } on \text{ } [0, 1/p) $$

The condition on $R$ (the number of prolates to be summed) is $R \geq 2 \pi \Omega T$ where $2 \pi \Omega T $ is the so-called "time-bandwidth product".

As an alternative (though difficult but equivalent) task one may prove that for: $$\lambda_{max}(\mathbf{S}) = \underset{u \in [0, 1/p)}{\mathrm{ess \ sup}} \ \underset{1\leq i \leq p}{\ \mathrm{max}\ } \lambda_i (\mathbf{S})(u)$$ and $$\lambda_{min}(\mathbf{S}) = \underset{u \in [0, 1/p)}{\mathrm{ess \ inf}} \ \underset{1\leq i \leq p}{\ \mathrm{min}\ } \lambda_i (\mathbf{S})(u)$$

we have $0< \lambda_{min}(\mathbf{S}) \leq \lambda_{max}(\mathbf{S}) < \infty$

I have performed numerical experiments and there seems to be ample evidence that the determinant is indeed positive. I have surveyed a lot of literature on the Prolates, but have not been able to make any progress. Any pointers/hints would be highly appreciated.

$\endgroup$
1
$\begingroup$

I have worked for some time on the problem and made some progress. The argument is basically inductive. First all I would like to tell that this problem arises in Gabor Theory if one wants to prove that a Gabor frame can be generated by multiple prolate spheroidal wave functions i.e. if one wants to prove that the sequence $${e^{2 \pi i m b} \varphi_r(t-n a): m, n, r \in \mathbb{Z}, 0 \leq r \leq R-1}$$ forms a "frame" (using what is known as the Zak Transform). So far this has not been done in the literature on prolates or Gabor theory.

The key is to take into consideration the well-known fact that the prolates form a Complete Chebyshev system on the interval $[-1, 1]$. So (assuming $T$ and $\Omega$ are both 1) the following determinant is non-zero: $$ \Delta \left( {\begin{array}{cccc} \varphi_0 & \varphi_1 & \cdots & \varphi_r\\ x_0 & x_1 & \cdots & x_r\\ \end{array} } \right)= \mathrm{det}(\varphi_i(x_j))_{i,j=0}^{n}, \text{ }\forall r \in \mathbb{Z_+}, 0 \leq x_0< x_1 <\cdots< x_r \leq 1. $$

Writing out explicitly, the condition is given as: $$\left| \begin{array}{cccc} \varphi_0(x_0) & \varphi_0(x_1) & \cdots & \varphi_0(x_r)\\ \varphi_1(x_0) & \varphi_1(x_1) & \cdots & \varphi_1(x_r)\\ \vdots & \vdots & & \vdots\\ \varphi_r(x_0) & \varphi_r(x_1) & \cdots & \varphi_r(x_r)\\ \end{array} \right| \neq 0$$ for any finite set $\{x_0, x_1, \ldots, x_r\} \subseteq [0,1]$.

I will give the simple argument for $p=2$ and $R \geq p$. I have even done it for $p=3$ but I need an elegant inductive argument for all $p$. Introduce the slight abuse of notation: $$ \varphi_r(i) \equiv \varphi_r\left(u+\frac{i}{p}\right), \text{ } i=0,1,\ldots, p-1 $$

For $R=2$ and $p=2$, $\mathrm{det}(\mathbf{S})$ reads: $$\left| \begin{array}{cccc} \varphi_0(0)\varphi_0(0)+\varphi_1(0)\varphi_1(0) & \varphi_0(0)\varphi_0(1)+\varphi_1(0)\varphi_1(1)\\ \varphi_0(1)\varphi_0(0)+\varphi_1(1)\varphi_1(0) & \varphi_0(1)\varphi_0(1)+\varphi_1(1)\varphi_1(1)\\ \end{array} \right|$$ This deteminant can be decomposed into four determinants as $$\left| \begin{array}{cc} \varphi_0(0)\varphi_0(0) & \varphi_0(0)\varphi_0(1) \\ \varphi_0(1)\varphi_0(0) & \varphi_0(1)\varphi_0(1) \end{array}\right| + \left| \begin{array}{cc} \varphi_1(0)\varphi_1(0) & \varphi_1(0)\varphi_1(1) \\ \varphi_1(1)\varphi_1(0) & \varphi_0(1)\varphi_1(1) \end{array}\right| + \left| \begin{array}{cc} \varphi_0(0)\varphi_0(0) & \varphi_1(0)\varphi_1(1) \\ \varphi_0(1)\varphi_0(0) & \varphi_1(1)\varphi_1(1) \end{array}\right| + \left| \begin{array}{cc} \varphi_1(0)\varphi_1(0) & \varphi_0(0)\varphi_0(1) \\ \varphi_1(1)\varphi_1(0) & \varphi_0(1)\varphi_0(1) \end{array}\right|$$

The first two determinants are easily seen to be zero, while the sum of the last two determinants is equal to (after taking out common factors in entries in a given column): $$\left| \begin{array}{cc} \varphi_0(0) & \varphi_0(1) \\ \varphi_1(0) & \varphi_1(1) \end{array}\right|^2$$

Which is positive by the Chebyshev assumption. An inductive argument can be invoked for $R>2$. What I would request is a general inductive argument for all $p$, i.e., assuming $\mathrm{det}(\mathbf{S}) > 0$ for all $R \geq p=2$ and for a certain $p>2$ prove the proposition for $p+1$.

$\endgroup$
  • $\begingroup$ I was finally able to solve the problem by using the Cauchy-Binnet formula for the determinant of the product of two rectangular matrices $\endgroup$ – Iconoclast May 11 '18 at 22:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.