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 $U_{2,1,ldots , U_{2,d,: d in {ol Z$ and a generating element $phi$. Each MRA yields a system ${cal V ={ U_1^k U_{2,1^{l_1cdots U_{2,d^{l_d psi_n : | : n=0,ldots,N-1, , k in {ol Z ,l in {ol Z^d $, where the $psi_n$ are related to $phi$. Necessary and sufficient conditions on $U_1$, $U_{2,1,dots , U_{2,d$, $phi$ and $psi_n$ are given, such that $V$ is a Riesz system or basis in $H$.

Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) (msc 41A65), General harmonic expansions, frames (msc 42C15), None of the above, but in MSC2010 section 46Cxx (msc 46C99)
CWI. Probability, Networks and Algorithms [PNA]

van Eijndhoven, S.J.L, & Oonincx, P.J. (1997). Frames, Riesz systems and MRA in Hilbert spaces. CWI. Probability, Networks and Algorithms [PNA]. CWI.