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 $H$.

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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.