Quadrature Mirror Filters and Loop Groups

In this paper we want to show, that the finite impulse response quadratic mirror filters (QMF) associated to a tower of grids Γ ⊂ H = ZZ can be identified with a right coset of the subgroup Fix(TΓ⊥ ,Map(T n → U(N) : poly) of the group of polynomial loops Map(T → 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. 1 On leave from CNRS Centre de Physique Théorique, Luminy, Case 907, F-13288 Marseille, e-mail: hols.at.marcptnx1.univ-mrs.fr 2 Technische Universität Berlin, FB Mathematik, Strasse des 17. Juni 135, 10623 Berlin Matthias Holschneider and Ulrich Pinkall 1