Uncertainty principles in harmonic analysis

AbstractLet ƒ∈L2(Rn), ‖;ƒ‖2 = 1. Generalizing the Heisenberg uncertainty principle, lower bounds for the variance of the measure ¦ƒ(x)¦2dx are established under assumptions on the Fourier transform \̂tf. If E is a subset of Rn of codimension one which is suitably uniformly distributed, and b is a parameter which measures the maximum distance from a point to E, then the estimate Var(|ƒ(x)|2 dx) ⩾ cb−2 holds provided ∝E ¦\̂tf¦2 dσ ⩽ c′b−1 for suitable constants c and c′. Examples of sets E are families of parallel hyperspaces or concentric spheres. Analogous results are established for normalized L2 functions on the sphere Sn. Here the lower bound is infy ∈ Sn∫Sn|sin d(x, y)|2 |ƒ(x)|2 dx ⩾ cb−2. and the hypothesis on the spherical harmonic expansion ƒ = ∑k = 0∞ ƒk is ∑k ϵ A ∥ƒk∥22 ⩽ c′b−1, where A is a set of integers with gaps at most b for both even and odd numbers