SU I (2, F [ z,1/z ]) for F A Subfield of C

Abstract

Let F be a subfield of C closed under complex conjugation, and denote by U(2, F[z, 1/z]) the multiplicative group of two-by-two unitary matrices over the ring F[z, 1/z] where lzl = 1. Let SU1(2, F[z, 1/z]) be the subgroup of such unitary matrices with determinant equal to the constant polynomial 1 and which are equal to the identity matrix upon evaluation at z = 1. The main result is the Unique Factorization Theorem for S U1 (2, F[z, 1 /z]), which expresses each element as an unique product of a minimal number of simple factors in SU1(2, F[z, 1/z]). Since these factors are free of relations, it follows that SU1(2, F[z, 1/z]) is a (nonabelian) free group generated by the factors. Besides having interesting algebraic and topological properties from the standpoint of pure mathematics, these unitary groups have application to the theory of Quadrature Mirror Filter banks and to the theory of with compact support especially in parameterizing various classes of QMF banks and wavelet families. After this manuscript was completed, the author's attention was drawn to reference [V], which derives (nonunique) factorizations of the groups U(d, R[z, 1/z]) and U(d, C[z, 1/z]) for d > 2. Reference [V], part of the extensive QMF literature, concentrates mainly on the application of these factorizations to design QMF banks for particular digital signal processing problems. The theory of wavelets with compact support provides both a generalization of and an analytical viewpoint of QMF theory. Wavelets with compact support were recently discovered by Ingrid Daubechies as described in [DI] and [D2].