Local analysis of frame multiresolution analysis with a general dilation matrix

A multivariate semi-orthogonal frame multiresolution analysis with a general integer dilation matrix and multiple scaling functions is considered. We first derive the formulas of the lengths of the initial (central) shift-invariant space V-0 and the next dilation space V-1, and, using these formulas, we then address the problem of the number of the elements of a wavelet set, that is, the length of the shift-invariant space W-0 := V-1 circle minus V-0. Finally, we show that there does not exist a 'genuine' frame multiresolution analysis for which V-0 and V-1 axe quasi-stable spaces satisfying the usual length condition.