Fourier Methods for Piecewise Hermite Bicubic Or- Thogonal Spline Collocation

| Matrix decomposition algorithms employing fast Fourier transforms were developed recently by the authors to solve the systems of linear algebraic equations that arise when piecewise Hermite bicubic orthogonal spline collocation (OSC) is applied to certain separable elliptic boundary value problems on a rectangle. In this paper, these algorithms are interpreted as Fourier methods in analogy with Fourier methods for nite dierence schemes. This interpretation not only simplies the presentation of the OSC matrix decomposition algorithms but provides a natural unifying framework that closely parallels the Fourier method for the continuous problem. Like their nite dierence counterparts, OSC Fourier methods are based on knowledge of the solution of an eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and the associated matrix representation of the eigenfunctions are presented for various choices of boundary conditions.