integral is the product of the first fourier coefficients

Question

let $f\in L^1$ and $g \in L^{\infty}$, 2pi periodic

show $\lim_{n\to \infty} \frac{1}{2\pi} \int_0^{2\pi} f(t) g(nt) dt = \hat{f}(0) \hat{g}(0)$

i tried using expressing the rhs as integrals, but i could not match the lhs

any suggestions?

Answer 1

Hints: You can subtract off a constant from $g(t)$ and assume $\int_0^{2\pi} g(t)\,dt = 0$. First prove it for smooth $f(t)$ using integration by parts and then approximate a general $f(t)$ in $L^1$ by smooth functions.