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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 $\mathbb{T}$ ?

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

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  • $\begingroup$ Thank you very much. A last question: do you think that the answer to the question is yes for abitrary abelian locally compact groups $G$ and $H$ such that $H\subset \widehat{G}$ instead of the groups $\mathbb{R}$ and $\mathbb{Z}$? $\endgroup$ – Zouba Oct 9 '11 at 11:57
  • $\begingroup$ @Zouba There is a canonical projection $P:\mathbb{R}\to\mathbb{T}\ $ here and $g(x)=\displaystyle\sum_{y=P^{-1}(x)}f(y)\ $. In the general case I don't know. Perhaps if one is a covering space for another this reasoning still will be correct. $\endgroup$ – Andrew Oct 9 '11 at 12:40

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