Banded matrices with banded inverses, II: Locally finite decomposition of spline spaces

Abstract

Given two function spacesV ₀,V ₁ with compactly supported basis functionsC _i, F_i, i∈Z, respectively, such thatC _i can be written as a finite linear combination of theF _i's, we study the problem of decomposingV ₁ into a direct sum ofV ₀ and some subspaceW ofV ₁ in such a way thatW is spanned by compactly supported functions and that eachF _i can be written as a finite linear combination of the basis functions inV ₀ andW. The problem of finding such locally finite decompositions is shown to be equivalent to solving certain matrix equations involving two-slanted matrices. These relations may be reinterpreted in terms of banded matrices possessing banded inverses. Our approach to solving the matrix equations is based on factorization techniques which work under certain conditions on minors. In particular, we apply these results to univariate splines with arbitrary knot sequences.