Fourier coefficients converging

Question

I'm thinking about this question, which has no answer yet despite being on a bounty and having 100+ views.

Maybe it would be easier to start by asking this: Let $g\in C_0^{\infty}(\mathbb{R})$ (infinitely differentiable with compact support), and say $g(x)=0$ for all $|x|\geq N$.

Consider $$\hat{g}(y)=\int_{-\infty}^\infty g(x)e^{-ixy}dx$$

and $$\hat{f}(n)=\dfrac{1}{2L}\int_{-L}^Lf(y)e^{-in\pi y/L}dy$$ for $L>N$. Suppose $f$ is defined by $f(x)=g(x)$ on $[-L,L]$, and then extending so that $f$ is periodic of period $2L$. What can we say about $\hat{f}(n)$ and $\hat{g}(y)$ as $L\rightarrow\infty$?

Answer 1

What can we say about $\hat f(n)$ as $L\to\infty$?

$\hat f(n)\to 0$ because $$|\hat f(n)| \le \frac{1}{2L} \int_{\mathbb R} |g(x)|\,dx$$

What can we say about $\hat g(y)$ as $L\to\infty$?

$\hat g(y)$ does not depend on $L$.