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.