Wavelet Transform of the Dilation Equation

CABRELLI, CARLOS A.; MOLTER, URSULA M.

JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS

Australian Mathematical Society

Lugar: Canberra; Año: 1996 vol. 37 p. 474 - 489

0334-2700

In this article we study the dilation equation f(x) = SUM c_h f(2x-h) in L^2(R) using awavelet approach. We see that the structure of Multiresolution Analysis adapts very wellto the study of scaling functions. The equation is reduced to an equation in a subspace ofL^2(R) of much lower resolution. This simpler equation is then "wavelet transformed" toobtain 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.