Norms Concerning Subdivision Sequences and Their Applications in Wavelets

Abstract

Let a:=( a(α)) α∈ Z s be a finitely supported sequence of r× r matrices and M be a dilation matrix. The subdivision sequence {( a n (α)) α∈ Z s : n∈ N } is defined by a 1= a and a n+1 (α)= ∑ β∈ Z s a n (β)a(α−Mβ), α∈ Z s , n∈ N. Let 1≤ p≤∞ and f=( f 1,…, f r ) T be a vector of compactly supported functions in L p ( R s ). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L p -norms of the combinations of the shifts of f with the subdivision sequence coefficients: ‖ ∑ α∈ Z s a n (α)f(x−α)‖ p . Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method.