Inverses for a class of banded matrices and applications to piecewise cubic approximation

We give a simple criterion for the invertibility of a class of banded matrices that arise in the approximation by piecewise cubic polynomials. We also give a formula for the inverse in terms of the powers of a 2 × 2 matrix. We present sample applications of these results to interpolation and eigenvalue problems. As a side result wee find that the Gaussian points are best.