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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterization of scaling functions in a multiresolution analysis


Authors: P. Cifuentes, K. S. Kazarian and A. San Antolín
Journal: Proc. Amer. Math. Soc. 133 (2005), 1013-1023
MSC (2000): Primary 42C15; Secondary 42C30
Published electronically: November 19, 2004
MathSciNet review: 2117202
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Abstract: We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map $A: \mathbb{R} ^n\rightarrow \mathbb{R} ^n$ such that $A(\mathbb{Z} ^n) \subset \mathbb{Z} ^n$ and all (complex) eigenvalues of $A$ have absolute value greater than $1.$ In the general case the conditions depend on the map $A.$ We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when $A$is a diagonal matrix with all numbers in the diagonal equal to $2.$ There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.


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Additional Information

P. Cifuentes
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: patricio.cifuentes@uam.es

K. S. Kazarian
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: kazaros.kazarian@uam.es

A. San Antolín
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: angel.sanantolin@uam.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07786-X
PII: S 0002-9939(04)07786-X
Keywords: Multiresolution analysis, scaling function, Fourier transform, approximate continuity
Received by editor(s): March 6, 2003
Received by editor(s) in revised form: June 19, 2003
Published electronically: November 19, 2004
Additional Notes: The first two authors were partially supported by BFM2001-0189
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.