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Existence of multiwavelets in $\mathbb{R} ^n$

Authors: Carlos A. Cabrelli and María Luisa Gordillo
Journal: Proc. Amer. Math. Soc. 130 (2002), 1413-1424
MSC (2000): Primary 42C40; Secondary 42C30
Published electronically: October 12, 2001
MathSciNet review: 1879965
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Abstract: For a $q$-regular Multiresolution Analysis of multiplicity $r$ with arbitrary dilation matrix $A$ for a general lattice $\Gamma$ in $\mathbb{R} ^n$, we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if $2r(m-1) \geq n$ where $m$ is the absolute value of the determinant of $A$, then these conditions are always met, and therefore an associated wavelet basis of $q$-regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix $A$ for a lattice $\Gamma$.

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Additional Information

Carlos A. Cabrelli
Affiliation: Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina, and CONICET, Argentina

María Luisa Gordillo
Affiliation: Departamento de Informática, F.C.E.F.y N., Universidad Nacional de San Juan, Avda. José Ignacio de la Roza y Meglioli (5400) San Juan, Argentina

Keywords: Multiresolution Analysis, dilation matrix, multiwavelets, non-separable wavelets, wavelets
Received by editor(s): June 23, 2000
Received by editor(s) in revised form: November 19, 2000
Published electronically: October 12, 2001
Additional Notes: The research of the authors is partially supported by grants UBACyT TW84, CONICET, PIP456/98 and BID-1201/OC-AR-PICT-03134
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society