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

 

Local bases for refinable spaces


Authors: Carlos Cabrelli, Sigrid B. Heineken and Ursula M. Molter
Journal: Proc. Amer. Math. Soc. 134 (2006), 1707-1718
MSC (2000): Primary 39A10, 42C40, 41A15
Published electronically: December 5, 2005
MathSciNet review: 2204283
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Abstract: We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.


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Additional Information

Carlos Cabrelli
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina – and – CONICET, Argentina
Email: cabrelli@dm.uba.ar

Sigrid B. Heineken
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina
Email: sheinek@dm.uba.ar

Ursula M. Molter
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina – and – CONICET, Argentina
Email: umolter@dm.uba.ar

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08192-X
PII: S 0002-9939(05)08192-X
Keywords: Homogeneous functions, shift-invariant spaces, accuracy, refinable functions
Received by editor(s): August 3, 2003
Received by editor(s) in revised form: January 11, 2005
Published electronically: December 5, 2005
Additional Notes: The research of the authors was partially supported by Grants CONICET, PIP456/98, and UBACyT X058 and X108. The authors also acknowledge partial support from the Guggenheim Foundation and the Fulbright Commission during the period in which part of this research was performed.
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.