Frames, Riesz systems and multiresolution analysis in Hilbert spaces

The concept of multiresolution analysis (MRA) is introduced for arbitrary separable Hilbert spaces H. It is put in the general terms of unitary operators U-1 and U-2.1,..., U-2,U-d, d is an element of Z and a generating element phi. Each MRA yields a system V = {(U1U2.1l1)-U-k... U(2,d)(ld)psi(n)\n = 0,..,N - 1; k is an element of Z, l is an element of Z(d)}, where the psi(n) are related to phi. Necessary and sufficient conditions on U-1, U-2,U-1,..., U-2,U-d, phi and psi(n) are given, by means of properties of matrix-valued functions on the unit circle, such that V is a Riesz system or Riesz basis in H.