The support of a function with thin spectrum

Volume 67 / 1994

Kathryn Hare Colloquium Mathematicum 67 (1994), 147-154 DOI: 10.4064/cm-67-1-147-154

Abstract

We prove that if $E ⊆ Ĝ$ does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty $S ⊆ G$ there exists a constant c > 0 such that $∥ f1_S ∥_2 ≥ c ∥ f ∥ _2$ for all $f ∈ L^2(G)$ whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.

Authors

  • Kathryn Hare

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