Consider the following question
Let $f$ be a function such that $\int_{-\infty}^{+\infty}|f(x)|^2 d x<\infty$. Prove that $\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0$ for all integers $k$ and $l$ with $k \neq l$ if and only if $$ \int_{-\infty}^{+\infty}|\widehat{f}(t)|^2 e^{-i m t} d t=0 \quad \text { for all integers } m \neq 0 $$ where $\widehat{f}$ is the Fourier transform of $f$.
I can't seem to do either direction. Here is what I have tried. Let us first assume the first integral. Then I tried rewriting the second expression in terms of multiple integrals, however, I am having an issue when I can and can't pull things through the integral. Similalry, with the converse, even if I allowed myself to pull things through the integral sign I can't seem to get the desired expression.
This is from a second year past paper that can be found at https://www.maths.cam.ac.uk/undergrad/pastpapers/files/2002/PaperIB_4.pdf with this beeng question 8H.
If more detailed is required on anything please let me know.
I would appreciate it if anybody could shine some light on this question.
Consider the following statements:
$(1)$ $\{f(x + k)\}_{k\in \mathbb{Z}}$ is an orthogonal set in $L^2(\mathbb{R})$
$(2)$ $\{ e^{-imt}\widehat{f}(x) \}_{m \in \mathbb{Z}}$ is an orthogonal set in $L^2(\mathbb{R})$.
It is easily seen that our task is to prove that $(1) \iff (2)$. But this follows from the fact that $\widehat{f(\cdot + k)} = e^{ik\xi}\hat{f}$ and Plancherel's theorem: $$ \langle f, g\rangle = \langle\hat{f}, \hat{g}\rangle, $$ that is, the Fourier transform sends an orthogonal set to an orthogonal set.