Hadamard triples generate self-affine spectral measures

TL;DR

This paper proves that self-affine measures generated by Hadamard triples are spectral, confirming a long-standing conjecture, and extends results to Fourier frames under relaxed conditions.

Contribution

It establishes that measures from Hadamard triples are spectral and provides conditions for Fourier frames when the Hadamard condition is relaxed.

Findings

Self-affine measures from Hadamard triples are spectral.

Confirmed a long-standing conjecture in fractal harmonic analysis.

Provided sufficient conditions for Fourier frames in relaxed settings.

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 ${\br}^d$ in the sense that the matrix $$ \frac{1}{\sqrt{|\det R|}}\left[e^{2\pi i \langle R^{-1}b,\ell\rangle}\right]_{\ell\in L,b\in B} $$ is unitary. We prove that the associated fractal self-affine measure $\mu = \mu(R,B)$ obtained by an infinite convolution of atomic measures $$ \mu(R,B) = \delta_{R^{-1} B}\ast\delta_{R^{-2}B}\ast\delta_{R^{-3}B}\ast... $$ is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in $L^2(\mu)$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.