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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Frame wavelets in subspaces of $L^2(\mathbb R^d)$


Authors: X. Dai, Y. Diao, Q. Gu and D. Han
Journal: Proc. Amer. Math. Soc. 130 (2002), 3259-3267
MSC (2000): Primary 42-XX, 47-XX
Published electronically: June 11, 2002
MathSciNet review: 1913005
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Abstract: Let $A$ be a $d\times d$ real expansive matrix. We characterize the reducing subspaces of $L^2(\mathbb{R}^d)$for $A$-dilation and the regular translation operators acting on $L^2 (\mathbb{R}^d).$We also characterize the Lebesgue measurable subsets $E$of $\mathbb{R}^d$ such that the function defined by inverse Fourier transform of $[1/(2\pi)^{d/2}]\chi_{E}$generates through the same $A$-dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular $L^2(\mathbb{R}^d)$-norm.


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Additional Information

X. Dai
Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223

Y. Diao
Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223

Q. Gu
Affiliation: Department of Mathematics, Beijing University, Beijing, People’s Republic of China
Address at time of publication: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China

D. Han
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Address at time of publication: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06498-5
PII: S 0002-9939(02)06498-5
Keywords: Normalized tight frame wavelet set, reducing subspace, connectivity
Received by editor(s): January 5, 2001
Received by editor(s) in revised form: February 26, 2001, and June 6, 2001
Published electronically: June 11, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society