Poisson's summation formula

Question

It is said that the Fourier transform $\hat{f}(\omega)$ of a function $f(t)$ and the Fourier transform $\hat{b}(\omega)$ of its samples $b(k)=f(t)|_{t=k}$ are related by Poisson's summation formula and it's given by

$$\hat{b}(\omega)=\displaystyle\sum_{k\in{\mathbb{Z}}}\hat{f}(\omega+2\pi{k})$$

where

$$\hat{b}(\omega)=\displaystyle\sum_{k\in\mathbb{Z}}b(k)e^{-i\omega k}, \omega\in{\mathbb{R}}.$$

I just fail to see why, is this equation even right?

Answer 1

One informal interpretation of the Poisson summation formula is that $$\hat{b}(\omega) = \sum_{k \in \mathbb Z}\hat{f}(\omega + 2\pi k)$$ is a periodic function of $\omega$ of period $2\pi$, and therefore representable as a Fourier series. The coefficients of this Fourier series are the $b(k)$. If you write out the integral formula for the Fourier coefficient and manipulate it a bit, then for reasonably well-behaved functions, you get the relationship to the function $f(t)$.

Answer 2

Here is a formal approach, taken from Rudin's book on real and complex analysis. Let $$ \varphi(t)=\frac{1}{2\pi} \int f(x)e^{-itx}\, dx $$ and $$ F(x) = \sum_{k=-\infty}^{+\infty} f(x+2k\pi). $$ Then $F$ is $2\pi$-periodic and its $n$th Fourier coefficient is $\varphi(n)$, hence $F(x)=\sum \varphi(n) e^{inx}$. In particular, $$ \sum_{k=-\infty}^{+\infty} f(2k\pi) = \sum_{n=-\infty}^{+\infty} \varphi(n). $$ More generally, $$ \sum_{k=-\infty}^{+\infty} f(k \beta) = \sum_{n=-\infty}^{+\infty} \varphi(n\alpha) $$ whenever $\alpha>0$, $\beta>0$, $\alpha \beta = 2\pi$.

Answer 3

Yes, your equation is right. If you want a (short) proof and some information about the Poisson Summation formula (PSF), you can look at the Wikipedia page. Another form this equation takes is when you set $\omega=0$; then you get the very symmetric $$\sum_{k\in\mathbb Z}\hat{f}(2\pi k)=\sum_{k\in\mathbb Z}b(k).$$ One interesting interpretation of this result is through the Selberg trace formula. Indeed, this trace formula says that, on the circle $S^1$, the trace of the Laplacian $-\Delta=-\frac{d^2}{dx^2}$ can be computed in two ways: the right-hand side of the PSF is just the sum of the eigenvalues, whereas the left-hand side caan be interpreted as the sum over the periodic orbits of the geodesic flow on the circle. In short, in this context, the PSF implies that the two ways to compute the trace of the Laplacian will give you the same result. Which is what you would expect.

Another application of the PSF is in proving the analytic continuation of the Riemann zeta function, and more generally, of Dirichlet L-functions.