Inverses of Band Matrices and Local Convergence of Spline Projections

It is shown that the size of the entries in the inverse of a band matrix can be bounded in terms of the norm of the matrix, the norm of its inverse and the bandwidth. In many cases this implies that the entries of the inverse decay to zero exponentially as they move away from the diagonal. These results are used to obtain local convergence theorems for some spline projections.