Let $f,g: \mathbb R \to \Bbb R$ be two $2\pi$ periodic functions.
(a) If $f$ is $C^{\infty}$ prove that for each $n$ there is $C_n \in \Bbb R$ such that$$\left|\hat{f}_k\right| \leq \frac{C_n}{\left|k\right|^n}\quad \forall k \in \Bbb Z \setminus \{0\}$$ where $\hat{f_k} = \frac{1}{2\pi}\int_0^{2\pi} f(t) e^{ikt} dt$ is the $k$-th Fourier coefficient of $f$.
(b) For $f \in C^{\infty}$ and $g \in L^{\infty}$ prove that $$\lim_{n\to \infty} \int_0^{2\pi} f(t) g(nt) dt = 2\pi \hat{f_0}\hat{g_0}$$
(c) Prove that (b) holds also if $f \in L^1$.
Now, I solved (a) using standard repeated integration by parts $n$ times, but what about the last two points? Can someone give me a hint? Thanks!
So, I think I understood how to do point (b). As suggested in the comments, first prove it for a function $f$ like $$f_m(t)= \sum_{\left| k\right|\leq m} \hat{f_k} e^{ikt}$$ Therefore, if $n \geq m$ we can write $$\int_0^{2\pi}f(t)g(nt) dt= \sum_{\left| k\right|\leq m} \hat{f_k} \int_0^{2\pi} e^{ikt}g(nt)dt = \sum_{\left| k\right|\leq m} \hat{f_k} \int_0^{2\pi} \sum_{l \in \Bbb Z}e^{ikt} e^{inlt}\hat{g}_l dt$$ Now all the terms are zero except the one with $l = k = 0$ so all that stuff is equal to $$2\pi\hat{f_0}\hat{g_0}$$ as wanted. So now for each $n$ we get
$$\int_0^{2\pi}f(t)g(nt) dt = \sum_ k \hat{f_k} \int_0^{2\pi} e^{ikt}g(nt)dt = \sum_ { k\geq n } \hat{f_k} \int_0^{2\pi} e^{ikt}g(nt)dt +\hat{f_0}\hat{g_0} 2\pi $$
Letting $n$ go to infinity we see that the first term in the last member goes to zero so we get the result.