Multiresolution Based on Weighted Averages of the Hat Function I: Linear Reconstruction Techniques

• Published in: SIAM journal on numerical analysis Vol. 36; no. 1; pp. 160 - 203
• Main Authors: Arandiga, Francesc, Donat, Rosa, Harten, Ami
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 1999
• Subjects: Approximation; Continuous functions; Decomposition; Exact sciences and technology; Interpolation; Linear transformations; Mathematical analysis; Mathematical functions; Mathematics; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Operator theory; Polynomials; Sciences and techniques of general use; Stencils; Weighted averages; Weighting functions; Reconstruction; Interpolation; Stability; Multiscale method; Prediction operator; Decomposition method; Refinement; Harten method; Reconstruction operator; Multirésolution technique; Computer aided design; Discretization method
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/s0036142996308770

In this paper we analyze a particular example of the general framework developed in [A. Harten, SIAM J. Numer. Anal., 33 (1996) pp. 1205-1256], the case in which the discretization operator is obtained by taking local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution. In Part I we use data-independent (linear) reconstruction techniques as our approximation tool. We show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations.