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Approximation by Smooth Multivariate Splines.

Author: C DE Boor; R DeVore; WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
Publisher: Ft. Belvoir Defense Technical Information Center JAN 1982.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:
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  Read more...
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Document Type: Book
All Authors / Contributors: C DE Boor; R DeVore; WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
OCLC Number: 227533149
Description: 23 pages

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.

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