A Fourier inversion problem

Question

Let's say I have a function $f\in L^1(\mathbb{Z})\cap L^2(\mathbb{Z}).$ Now using Plancherel I can get a function $g\in L^2(S^1)$ such that $\hat{g}=f$. Now I want to say $g\in L^1(S^1)$ so that I can apply Fourier inversion. Is there a way to show that or it's not always true?

Answer 1

Yes, since $S^1$ is a bounded set, the Cauchy-Schwarz inequality gives you $$ ∫_{S^1} |g| ≤ |S^1|^{1/2} \,\|g\|_{L^2(S^1)}. $$