Local uncertainty inequalities for Fourier series

Abstract

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