Convolution $f * g$

Question

Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$

I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?

Answer 1

$$(f*g)(x) = \int_{-\infty}^{\infty} f(t) e^{2\pi i(x-t)} dt $$

You can simplify the above as

$$(f*g)(x) = e^{2\pi i x} F(2\pi) $$

Where $F(w) $ is the Fourier transform of $f$

$$ F(w) = \int_{-\infty}^{\infty} f(t) e^{-iwt} dt $$