Hadamard triples generate self-affine spectral measures

Abstract

Let $R$ be an expanding matrix with integer entries and let $B,L$ be finite integer digit sets so that $(R,B,L)$ form a Hadamard triple on ${\br}^d$. We prove that the associated self-affine measure $\mu =\mu (R,B)$ is a spectral measure, which means it admits an orthonormal bases of exponential functions in $L^2(\mu )$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors.

References in corpus (3)

• Complex Hadamard matrices and the Spectral Set Conjecture
• Iterated function systems, Ruelle operators, and invariant projective measures
• Continuous and Discrete Fourier Frames for Fractal Measures

Cited by in corpus (5)

• Spectral property of self-affine measures on ${\mathbb{R}}^n$
• Fourier bases and Fourier frames on self-affine measures
• The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets
• Non-spectral problem for the planar self-affine measures
• Translational absolute continuity and Fourier frames on a sum of singular measures