Holes in the spectrum of functions generating affine systems

Abstract

Given a d x d expansive dilation matrix D, a measurable set E C R d is called a D t -dilation generator of R d if R d is tiled (modulo null sets) by the collection {(D t ) j E, j ∈ 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 R d .