Holes in the spectrum of functions generating affine systems

Abstract:

Given a $d\times d$ expansive dilation matrix $D$, a measurable set $E\subset \mathbb {R}^d$ is called a $D^t$-dilation generator of $\mathbb {R}^d$ if $\mathbb {R}^d$ is tiled (modulo null sets) by the collection $\{ (D^t)^j E, j\in \mathbb {Z}\}$. Our main goal in this paper is to prove certain results relating the support of the Fourier transform of functions generating a wavelet or orthonormal affine system associated with the dilation $D$ to an arbitrary set $E$ which is a $D^t$-dilation generator of $\mathbb {R}^d$.