BANDED MATRICES WITH BANDED INVERSES .2. LOCALLY FINITE DECOMPOSITION OF SPLINE SPACES

Given two function spaces V0, V1 with compactly supported basis functions C(i), F(i), i is-an-element-of Z, respectively, such that C(i) can be written as a finite linear combination of the F(i)'s, we study the problem of decomposing V1 into a direct sum of V0 and some subspace W of V1 in such a way that W is spanned by compactly supported functions and that each F(i) can be written as a finite linear combination of the basis functions in V0 and W. 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.