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Wavelet transform of the dilation equation

Published online by Cambridge University Press:  17 February 2009

Carlos A. Cabrelli  and
Ursula M. Molter
Ursula M. Molter
Affiliation:
Dept de Matemática, Universidad de Buenos Aires, Pabellón I, 1428 Capital Federal, Argentina.
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Abstract

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In this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 4 , April 1996 , pp. 474 - 489
Copyright
Copyright © Australian Mathematical Society 1996

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