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 ${\br}^d$ in the sense that the matrix $$\frac{1}{\sqrt{|\det R|}}{\left[e^{2\pi i⟨R^{-1}b,\ell ⟩}\right]}_{\ell \in L,b\in B}$$ 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.

arXiv commentarXiv admin note: text overlap with arXiv:1502.03209, arXiv:1506.01503, arXiv:1602.04750