Frame wavelets in subspaces of $L^2(\mathbb R^d)$
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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.References
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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
- MR Author ID: 356341
- 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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3259-3267
- MSC (2000): Primary 42-XX, 47-XX
- DOI: https://doi.org/10.1090/S0002-9939-02-06498-5
- MathSciNet review: 1913005