Extension of Bessel sequences to oblique dual frame sequences and the minimal projection

An extension of two Bessel sequences to oblique dual frame sequences and its applications to shift-invariant spaces are considered. The best-known situation where this kind of extension is necessary is the construction of a pair of biorthogonal multiresolution analyses, where two generating sets whose shifts are only assumed to be Bessel sequences are given. This extension naturally leads to consideration of the ‘minimal projection’ extending two closed subspaces. The existence or non-existence of the minimal projection is discussed.