Accession Number : ADA114506

Title :   Approximation by Smooth Multivariate Splines.

Descriptive Note : Technical summary rept.,

Corporate Author : WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER

Personal Author(s) : DE Boor,C ; DeVore,R

PDF Url : ADA114506

Report Date : Jan 1982

Pagination or Media Count : 23

Abstract : One of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh. This report shows this to be untrue in the multivariate setting. In particular, it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant. In an extreme case, piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or = RHo continuous will fail to approximate certain smooth functions at all (as the grid goes to zero) unless RHo is kept below (r-3)/2. During the analysis of approximation on a certain regular triangular grid, a novel kind of bivariate B-spline is introduced. This B-spline, in contrast to the established multivariate B-spline derived from a simplex, can be made to have all its breaklines in a given regular grid. This makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation, and its properties will be explored further.

Descriptors :   *Splines(Geometry), *Multivariate analysis, Polynomials, Approximation(Mathematics), Variations, Grids, Geometry, Finite element analysis, Discontinuities, Differential equations

Subject Categories : Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE