Frame multiresolution analysis (Q2706507)

Define two operators on \(L^2(R)\) by \((Df)(x)=2^{1/2}f(2x), \;\;(Tf)(x)=f(x-1)\). In this paper, a frame multiresolution analysis is defined to consist of a family of bounded operators \(\{A_j\}_{j\in Z}\) on \(L^2(R)\) and a function \(\phi \in L^2(R)\), for which (let \(V_j=\text{ran} A_j\) and \(Z_j=\text{ker} A_j\)) (i) \(A_j^2=A_j\), \ (ii) \( V_j\subset V_{j+1}, \;Z_{j+1}\subset Z_j\), (iii) \(DV_j=V_{J+1}, DZ_{j}=Z_{j+1}\), (iv) \(TA_0=A_0T\), \ (v) \(\{T^n\phi\}\) is a frame for \(V_0\), \ (vi) \(\overline{\cup}V_j= L^2(R), \;\cap V_j= \{ 0\}\). A FMRA generalizes Feauveau's model for a non-orthogonal multiresolution analysis (Feauveau has a stronger condition than (v)). It is proved that one can associate a dual FMRA to each FMRA, and that most of the properties for Feauveaus' model also hold for FMRA's. Sufficient conditions for the existence of a function \(\psi\) such that \(\{T^n\psi\}_{n\in Z}\) is a frame for the orthogonal complement of \(V_{j+1}\) in \(V_j\) and \(\{D^jT^n\psi\}_{j,n\in Z}\) is a frame for \(L^2(R)\) are given.