Frame wavelets in subspaces of

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

DOI:
https://doi.org/10.1090/S0002-9939-02-06498-5

Published electronically:
June 11, 2002

MathSciNet review:
1913005

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real expansive matrix. We characterize the reducing subspaces of for -dilation and the regular translation operators acting on We also characterize the Lebesgue measurable subsets of such that the function defined by inverse Fourier transform of generates through the same -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 -norm.

**1.**X. Dai, Y. Diao and Q. Gu,*Normalized Tight Frame Wavelet Sets,*Proc. Amer. Math. Soc., to appear.**2.**X. Dai and D. Larson,*Wandering vectors for unitary systems and orthogonal wavelets,*Memoirs Amer. Math. Soc., 134(1998), no. 640. MR**98m:47067****3.**X. Dai, D. Larson and D. Speegle,*Wavelet sets in**,*J. Fourier Anal. Appl., 3(1997), 451-456. MR**98m:42048****4.**X. Dai and S. Lu,*Wavelets in subspaces,*Mich. J. Math., 43(1996), 81-98. MR**97m:42021****5.**N. Dunford and J. Schwartz,*Linear Operators*Part I, Wiley-Interscience, 1958. MR**22:8302****6.**D. Han and D. Larson,*Bases, Frames and Group representations,*Memoirs. Amer. Math. Soc., 147(2000). MR**2001a:47013****7.**D. Speegle,*The**-elementary wavelets are path-connected,*Proc. Amer. Math. Soc., 127(1999), 223-233. MR**99h:42045**

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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:
https://doi.org/10.1090/S0002-9939-02-06498-5

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