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Holes in the spectrum of functions generating affine systems

Authors:	Jean-Pierre Gabardo Yun-Zhang Li

Institution:	Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 ; Department of Applied Mathematics, Beijing University of Technology, Beijing, 100022, People's Republic of China

Abstract:	Given a $d\times d$ expansive dilation matrix , a measurable set $E\subset \mathbb{R}^d$ is called a -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 to an arbitrary set which is a -dilation generator of $\mathbb{R}^d$ .

Keywords:	Affine systems wavelets dilation generator

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