Wavelet transform of the dilation equation
In this article we study the dilation equation f(x) = ∑hchf(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...
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| Autores principales: | , |
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1996
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03342700_v37_n4_p474_Cabrelli http://hdl.handle.net/20.500.12110/paper_03342700_v37_n4_p474_Cabrelli |
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| Sumario: | In this article we study the dilation equation f(x) = ∑hchf(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. © Australian Mathematical Society, 1996. |
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