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Holes in the spectrum of functions generating affine systems

Authors: Jean-Pierre Gabardo and Yun-Zhang Li
Journal: Proc. Amer. Math. Soc. 135 (2007), 1775-1784
MSC (2000): Primary 42C40; Secondary 42C15
Published electronically: November 7, 2006
MathSciNet review: 2286088
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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$.

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Additional Information

Jean-Pierre Gabardo
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Yun-Zhang Li
Affiliation: Department of Applied Mathematics, Beijing University of Technology, Beijing, 100022, People’s Republic of China

Keywords: Affine systems, wavelets, dilation generator
Received by editor(s): September 8, 2005
Received by editor(s) in revised form: February 2, 2006
Published electronically: November 7, 2006
Additional Notes: The first author was supported by an NSERC grant.
The second author was supported by the Natural Science Foundation of Beijing and the Foundation of Educational Ministry of China
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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