Limits of an Improper Integral: $\lim\limits_{n\to\infty} \int_{-\infty}^{+\infty}f(x)\sin(nx)dx$

Question

Is

$$ \lim\limits_{n\to\infty} \int_{-\infty}^{+\infty}f(x)\sin(nx)dx=0 $$

when $\int_{-\infty}^{+\infty}f(x)dx$ is convergent but not absolutely convergent?

Answer 1

Try the inverse Fourier transform of $$\hat{f}(\omega)= \sum_{k\ge 1} e^{-(k^2 (\omega-k))^2}e^{i \omega e^k}\in L^1\cap L^2$$

We have $\lim_{n\to \infty}\hat{f}(n)e^{-i n e^k}=1,\lim_{n\to -\infty}\hat{f}(n)=0$.

The $e^{i \omega e^k}$ term is there to ensure that $\lim_{A\to \infty}\int_{-A}^A e^{-inx} f(x)dx$ converges to $\hat{f}(n)$.