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Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

Authors: Qing Gu and Deguang Han
Journal: Proc. Amer. Math. Soc. 133 (2005), 815-825
MSC (2000): Primary 42C15, 47B38
Published electronically: September 29, 2004
MathSciNet review: 2113932
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Abstract: We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix $A$ and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

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Additional Information

Qing Gu
Affiliation: Department of Mathematics, East China Normal University, Shanghai, Peoples Republic of China

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Keywords: Wavelet, wavelet frame, frame multiresolution analysis, shift-invariant subspace, dimension function
Received by editor(s): February 25, 2002
Received by editor(s) in revised form: November 11, 2003
Published electronically: September 29, 2004
Additional Notes: This paper is a revised version based on an earlier circulated preprint: “Translation invariant subspaces and general multiresolution analysis", 1999.
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

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