Periodicity of the spectrum in dimension one

Abstract

A bounded measurable set , of Lebesgue measure 1, in the real line is called spectral if there is a set ƒ of real numbers ("frequencies") such that the exponential functions e .x/D exp.2 i x/, 2 ƒ, form a complete orthonormal system of L 2 ./.Such a set ƒ is called a spectrum of .In this note we prove that any spectrum ƒ of a bounded measurable set  ‫ޒ‬ must be periodic.

Locations

• Analysis & PDE - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF