Define a function h(x)=∑k=−∞∞f(x+2kπ), assuming that the series converges to a function satisfying Dirichlet conditions (Chapter 7, Section 6). Verify that h(x) does have period 2π.
(a) Expand h(x) in an exponential Fourier series h(x)=∑−∞∞cneinx; show that cn=g(n) where g(α) is the Fourier transform of f(x). Hint: Write cn as an integral from 0 to 2π and make the change of variable u=x+2kπ. Note that e−2inkπ=1, and the sum on k gives a single integral from −∞ to ∞.
(b) Let x=0 in (a) to get Poisson's summation formula ∑−∞∞f(2kπ)=∑−∞∞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.)