Complex Methods | Part IB, 2002

Let ff be a function such that +f(x)2dx<\int_{-\infty}^{+\infty}|f(x)|^{2} d x<\infty. Prove that

+f(x+k)f(x+l)dx=0 for all integers k and l with kl,\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0 \quad \text { for all integers } k \text { and } l \text { with } k \neq l,

if and only if

+f^(t)2eimtdt=0 for all integers m0\int_{-\infty}^{+\infty}|\widehat{f}(t)|^{2} e^{-i m t} d t=0 \quad \text { for all integers } m \neq 0

where f^\widehat{f} is the Fourier transform of ff.

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