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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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Published in: | Applied and computational harmonic analysis 2005-11, Vol.19 (3), p.386-431 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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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. |
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ISSN: | 1063-5203 1096-603X |
DOI: | 10.1016/j.acha.2004.12.006 |