article · 1995
Two-scale dilation equations and the cascade algorithm
Cited by 32Abstract and details only
Abstract
We study the two--scaled dilation equation f(x) = n X k=0 c k f(2x \\Gamma k) where the coefficients c k are real and P k c 2k = P k c 2k+1 = 1. By expressing the dilation equation in matrix product form we prove a necessary and sufficient condition for the cascade algorithm (introduced by Daubechies and Lagarias) to converge uniformly to a continuous solution. We also establish several basic relations between the convergence of infinite products of matrices and the existence and regularity of solutions to the two--scale dilation equations. INTRODUCTION Two--scale dilation equations arise in many diverse applications. The general equation is f(x) = N X k=0 c k f(2x \\Gamma k) (1) where X k even c k = X k odd c k = 1: Often equation (1) has compactly supported solutions supp(f) ` [0; N ], and we usually seek the normalized one Z f(x)dx = 1: If c k = \\Gamma N k \\Delta . 2 N \\Gamma1 then f is the normalized B--spline of degree N \\Gamma 1, given by f(x) = (\\Gamma1...
Cite this paper
Wang, Y. (1995). Two-scale dilation equations and the cascade algorithm.
Wang, Yang. “Two-scale dilation equations and the cascade algorithm.” 1995.
Wang, Yang. 1995. “Two-scale dilation equations and the cascade algorithm.”
Wang, Y. (1995) ‘Two-scale dilation equations and the cascade algorithm’.
@article{wang1995scale,
title = {{Two-scale dilation equations and the cascade algorithm}},
author = {Wang, Yang},
year = {1995}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.