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.



Determinant of a matrix involving the Prolate Spheroidal Wave Functions
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"](https://en.wikipedia.org/wiki/Frame_(linear_algebra)) (using what is known as the [Zak Transform](https://en.wikipedia.org/wiki/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 $p+1$. 


