Frames, Riesz systems and multiresolution analysis in Hilbert spaces

AbstractThe 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 ν = 1kU2,1l1 … U2,dldψn ¦ n = 0,…, N − 1, k e Z, l e Zd, where the ψn are related to f. Necessary and sufficient conditions on U1, U2,1,…, U2,d, f and ψn 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