Suppose that $f \in L^p(\mathbb R), 1 \leq p \leq 2$ is such that the Fourier transform of $f$ has support in the interval $[0,1]$. If $p=2$ then by Shannon's sampling Theorem we have the for every $a \in (0,1]$ the implication $$ f(an) = 0 \ \ \forall n \in \mathbb Z \implies f =0. $$ Does this implication holds also holds for $p=1$ and every $a \in (0,1]$?
2026-04-06 13:05:47.1775480747
Shannon sampling for $L^1(\mathbb R)$ functions
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