Uncertainty principles in harmonic analysis (Q1824143)

Suppose E is a subset of \({\mathbb{R}}^ n\) of codimension one which is ``uniformly distributed''. It is shown that if \(f\in L^ 2({\mathbb{R}}^ n)\), \(\| f\|_ 2=1\), \(Var(f)<\infty\), and \(\int_{E}| \hat f(\sigma)|^ 2d\sigma \leq C_ 1\) (where \(d\sigma\) is surface measure) then \(Var(f)\geq C_ 2(E,C_ 1)\). ``Uniformly distributed'' sets include parallel hyperplanes and concentric spheres which are uniformly spaced. A computation with the sphere suggests similar results for compact Riemannian manifolds.