Computational aspects of spline-wavelets

This dissertation deals with two topics on spline-wavelets with minimal support. Part one. Fast block-realtime algorithms for cardinal spline interpolation and cardinal spline-wavelet decomposition suitable for large data sets are presented. In order to find an interpolating cardinal spline on the real line, the given sequence of input values is partitioned into blocks of finite length. The interpolation algorithm is based on the factorization of the Euler-Frobenius polynomial, and on forward/backward substitution applied to one block at a time. The spline-wavelet decomposition is viewed as a cardinal spline interpolation problem of even order, and the interpolation algorithm is applied. An error analysis demonstrates the stability of the algorithm. The decomposition algorithm has only 2 to 3 times the numerical complexity of the reconstruction, and hence is very efficient. Numerical examples illustrate the algorithms.; Part two. Minimally supported spline-wavelets with multiple non-uniform knots on a bounded interval are considered. General knot refinements of arbitrarily many new knots between two old knots are allowed. A spline-wavelet of order m with minimal support is constructed as the m-th derivative of a locally supported spline of order 2m satisfying certain Hermite-interpolation conditions. The interpolation problem leads to a system of linear equations with a positive definite, skew-banded coefficient matrix to which LU-factorization can be applied without destroying its zero-entry structure. This provides an efficient method for the construction of a minimally supported spline-wavelet basis. A decomposition algorithm is developed based on Hermite-interpolation. Knot insertion is used for the reconstruction. The algorithms are tested and numerical results are included.