Quadrature Mirror Filters and Loop Groups

Abstract

In this paper we want to show, that the finite impulse response quadratic mirror filters (QMF) associated to a tower of grids $Γ\subset H=\bf Z^n$ can be identified with a right coset of the subgroup Fix$(T_{Γ^{\perp}}$,Map$ ({\bf T}^n\to U(N)$: poly) of the group of polynomial loops Map$({\bf T}^n\to U(N)$: poly) with $N=|H/Γ|$. The QMF with some vanishing moments can be identified with cosets of subgroups. The problem to parameterize all finite impulse response QMF in arbitrary space dimensions is now equivalent to factorize all polynomial loops.