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 U1 and U2.1, …, U2.d, d e Z and a generating element f. Each MRA yields a system ¿ = 1kU2,1l1 … U2,dld¿n ¦ N = 0,…, N - 1, k e Z, l e Zd, where the ¿n are related to f. Necessary and sufficient conditions on U1, U2,1,…, U2,d, f and ¿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