Fourier transform Suppose $1< p<\infty$. Let $f$ be a continuous function with compact support defined on $\mathbb{R}$. Does it exist a function $g \in L^p(\mathbb{T})$ such that: 
$$
\widehat{f}|_{\mathbb{Z}}=\widehat{g}
$$ 
where $\widehat{f}$ denote the Fourier transform on $\mathbb{R}$ and $\widehat{g}$ the Fourier transform on $\mathbb{T}$ ? 

Taking
$$
g(x)=\sum_{n=-\infty}^\infty f(x+2\pi n)
$$
will give the desired result since $g\in C(\mathbb{T})$.
