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Parseval frame wavelets with E n ( 2 ) -dilations

We study Parseval frame wavelets in L 2 ( R n ) with matrix dilations of the form ( D f ) ( x ) = 2 f ( A x ) , where A is an arbitrary expanding n × n matrix with integer coefficients, such that | det A | = 2 . We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wave...

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Bibliographic Details
Published in:Applied and computational harmonic analysis 2005-11, Vol.19 (3), p.386-431
Main Authors: Bakić, Damir, Krishtal, Ilya, Wilson, Edward N.
Format: Article
Language:English
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Summary:We study Parseval frame wavelets in L 2 ( R n ) with matrix dilations of the form ( D f ) ( x ) = 2 f ( A x ) , where A is an arbitrary expanding n × n matrix with integer coefficients, such that | det A | = 2 . We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2004.12.006