Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T05:27:24.219Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Berger, M. A. and Wang, Y., “Bounded semigroups of matrices”, Linear Alg. Appl. 166 (1992) 21–27.CrossRefGoogle Scholar
[2]Berger, M. A. and Wang, Y., “Multi-scale dilation equations and iterated function systems”, to appear.Google Scholar
[3]Coifman, R., Meyer, Y. and Wickerhauser, M. V., “Wavelet analysis and signal processing”, in Wavelets and Their Applications (ed. Ruskai, M. B. et al. ), (Jones and Bartlett, Boston, 1992) 153–178.Google Scholar
[4]Coifman, R., Meyer, Y. and Wickerhauser, M. V., “Size properties of wavelet packets”, in Wavelets and Their Applications (ed. Ruskai, M. B. et al. ), (Jones and Bartlett, Boston, 1992) 453–470.Google Scholar
[5]Colella, D. and Heil, C., “Characterizations of scaling functions, I. Continuous solutions”, SIAM J. Matrix Anal. Appl. 15 (1994) 496–518.CrossRefGoogle Scholar
[6]Daubechies, I., “Orthonormal bases of compactly supported wavelets”, Coram. Pure Appl. Math. 41 (1988) 909–996.CrossRefGoogle Scholar
[7]Daubechies, I., “Ten lectures on wavelets”, CBMS-NSF Series in Applied Mathematics 61 (SIAM Publications, Philadelphia, 1992).Google Scholar
[8]Daubechies, I. and Lagarias, J. C., “Two-scale difference equations I. Existence and global regularity of solutions”, SIAM J. Math. Anal. 22 (1991) 1388–1410.CrossRefGoogle Scholar
[9]Daubechies, I. and Lagarias, J. C., “Sets of matrices all infinite products of which converge”, Linear Alg. Appl. 161 (1992) 227–263.CrossRefGoogle Scholar
[10]Daubechies, I. and Lagarias, J. C., “Two-scale difference equations n. Infinite products of matrices and fractals”, SIAM J. Math. Anal. 23 (1992) 1031–1079.CrossRefGoogle Scholar
[11]Heil, C. and Strang, G., “Continuity of the joint spectral radius”, preprint, 1993.Google Scholar
[12]Lagarias, J. C. and Wang, Y., “The finitness conjecture for the generalized spectral radius of a set of matrices”, preprint, 1992.Google Scholar
[13]Mallat, S., “Multiresolution approximations and wavelet orthonormal basis of L2(R)”, Trans. Amer. Math. Soc. 315 (1989) 69–87.Google Scholar
[14]Meyer, I., Ondelettes, fonctions splines et analyses graduees (Lectures given at the University of Torino, Italy, 1986).Google Scholar
[15]Meyer, I., Ondelettes et operateurs I (Hermann, 1988).Google Scholar
[16]Micchelli, C. A. and Prautzsch, H., “Uniform refinement of curves”, Linear Alg. Appl. 114/115 (1989) 841–870.CrossRefGoogle Scholar
[17]Rota, G. C. and Strang, G., “A note on the joint spectral radius”, Indagationes Mathematicae 22 (1960) 379–381.CrossRefGoogle Scholar
[18]Wang, Y., “On two-scale dilation equations”, to appear.Google Scholar