Local uncertainty inequalities for Fourier series

Abstract:

Necessary and sufficient conditions are given on $\alpha$, $\beta$ and $t$ for there to exist a constant $K$ such that \[ {\left ( {{{\sum \limits _{n \in E} {\left | {\hat f(n)} \right |} }^2}} \right )^{1/2}} \leqslant K{\left | E \right |^\alpha }{\left \| {f{{\left | x \right |}^\beta }} \right \|_t}\] for all $f \in {L^1}({T^d})$ and finite $E \subset {Z^d}$.