Spectrality and tiling by cylindric domains

Abstract

A bounded set $\Omega \subset {\mathbb{R}}^d$ is called a spectral set if the space $L^2(\Omega )$ admits a complete orthogonal system of exponential functions. We prove that a cylindric set $\Omega$ is spectral if and only if its base is a spectral set. A similar characterization is obtained of the cylindric sets which can tile the space by translations.

References in corpus (2)

• Fuglede's spectral set conjecture for convex polytopes
• Tilings by translation

Cited by in corpus (6)

• The Fuglede conjecture for convex domains is true in all dimensions
• Fuglede's spectral set conjecture for convex polytopes
• The structure of translational tilings in ${\mathbb{Z}}^d$
• Spectral sets and weak tiling
• Riesz bases of exponentials and multi-tiling in finite abelian groups
• On product spectral sets and functional tiles in ${\mathbb{Q}}_p^d$