Chapter 7.13, Problem 21MP

Define a function $h(x)={\displaystyle \underset{k=-\infty }{\overset{\infty }{\sum }}f}(x+2k\pi ),$ assuming that the series converges to a function satisfying Dirichlet conditions (Section 6). Verify that $h(x)$ does have period $2\pi$.

(a) Expand $h(x)$ in an exponential Fourier series $h(x)={\displaystyle \underset{-\infty }{\overset{\infty }{\sum }}{c}_n}{e}^{inx};$ show that $c_n=g(n)$ where $g(\alpha )$ is the Fourier transform of $f(x).$ Hint: Write $c_n$ as an integral from 0 to 2 $\pi$ and make the change of variable $u=x+2k\pi .$ Note that $e^{-2ink\pi }=1,$ and the sum on $k$ gives a single integral from $-\infty$ to $\infty$.

(b) Let $x=0$ in (a) to get Poisson’s summation formula ${\displaystyle \underset{-\infty }{\overset{\infty }{\sum }}f}(2k\pi )={\displaystyle \underset{-\infty }{\overset{\infty }{\sum }}g}(n)$.

This result has many applications; for example: statistical mechanics, communication theory, theory of optical instruments, scattering of light in a liquid, and so on. (See Problem 22.)