Background
The Weyl-Heisenberg set $\mathcal{S}$ of sinc functions forms an orthonormal basis for $L_2(R)$, where we have
$$\mathcal{S}=\{s_{km}(t)=\frac{sin(\pi (t-m)}{\pi (t-m)}\exp(j2\pi kt)|m,k\in\mathbb{Z}\}$$
i.e. $\mathcal{S}$ contains time- and frequency shifted versions of the sinc. For example, two elements for different $k$ are orthogonal to each other:
$$ \left<s_{k,m},s_{k+1,m}\right>=\left<\mathcal{F}\{s_{k,m}(t)\},\mathcal{F}\{s_{k+1,m}(t)\}\right>=0 $$
where $\mathcal{F}$ is the Fourier Transform. We can directly see this from the fact that $\mathcal{F}\{s_{km}(t)\}(f)$ is a Rectangular function in $f$ of width $1$. Then, when multiplying to different frequency shifts of the $s_{k,m}(t)$ function, the rects do not overlap (or only overlap at a set of Lebesgue measure 0 (i.e. at the discontinuity of the rects), and hence the scalar product is 0.
Problem
Consider the periodic sinc $\tilde{s}_{k,m}(t)=\sum_{l\in\mathbb{Z}}s_{k,m}(t-lM)$. The WH-Set of periodic sincs forms an orthogonal basis for $L_2([0,M])$. Since it's periodic, it's Fourier Transform is discrete (i.e. consists of Dirac impulses).
- What happens, if the Dirac impulse occurs at the discontinuity of the rectangular spectrum?
- We can define the value at the discontinuity to be 1/2. But then, it seems the orthogonality between two frequency-shifted functions is lost, because the Diracs at the discontinuity overlap. And the integral over the frequency domain would not be zero.
- We can define the rectangular function to be 1 at one side of the rectangle, but 0 at the other side. Then, orthogonality is given, but the corresponding time-domain function would be complex (not symmetric in frequency domain).
How can this paradox be resolved? I have the feeling that it relates to the problem that the periodization actually does not converge in the common sense, but I'm not enough into these delicate mathematical problems.