
Accession Number : ADA118606
Title : Approximation Order from Bivariate C1Cubics: A Counter Example.
Descriptive Note : Technical summary rept.,
Corporate Author : WISCONSIN UNIVMADISON MATHEMATICS RESEARCH CENTER
Personal Author(s) : DE Boor,C ; Hoellig,K
PDF Url : ADA118606
Report Date : Jun 1982
Pagination or Media Count : 14
Abstract : It is shown that the space of bivariate C1 piecewise cubic functions on a hexagonal mesh of size h approximates to certain smooth functions only to within O(h3) even though it contains a local partition of every cubic polynomial. One measures the approximation power of a family S of piecewise polynomial approximating functions on some partition in terms of the meshsize h of that partition. Typically, the error of approximation goes to zero like hr as the meshsize goes to zero, with r depending on the smoothness of the function being approximated. There is a maximal r typical for the space S used, and faster convergence rates are possible only for very special functions. Naturally, one would like this optimal rate or approximation order hr to be as fast as possible, i.e., would like the maximal r to be as large as possible. In any case, it is an important practical question to ascertain, for a given approximating space S, what its optimal approximation order is.
Descriptors : *Approximation(Mathematics), *Bivariate analysis, Cubic spline technique, Order statistics, Polynomials, Mesh, Optimization, Convergence, Special functions(Mathematical)
Subject Categories : Statistics and Probability
Distribution Statement : APPROVED FOR PUBLIC RELEASE