Increasing the polynomial reproduction of a quasi-interpolation operator

JOURNAL OF APPROXIMATION THEORY

卷:161

Article

Waldron, Shayne

关键词: Quasi-interpolation;    Lagrange interpolations;    Bernstein polynomial;    Finite element method;    Multivariate polynomial approximation;    Error formula;    Multipoint Taylor formula;    Divided differences;    Chu-Vandermonde convolution;   

DOI  :  10.1016/j.jat.2008.08.011

来源: Elsevier

【 摘 要 】

Quasi-interpolation is an important tool, used both in theory and in practice, for the approximation of smooth functions from univariate or multivariate spaces which contain Pi(m) = Pi(m) (R(d)), the d-variate polynomials of degree <= in. In particular, the reproduction of Pi(m) leads to an approximation order of m + 1. Prominent examples include Lagrange and Bernstein type approximations by polynomials, the orthogonal projection onto Pi(m) for some inner product, finite element methods of precision in, and multivariate spline approximations based on macroelements or the translates of a single spline. For such a quasi-interpolation operator L which reproduces Pi(m) (R(d)) and any r >= 0, we give an explicit construction of a quasi-interpolant R(m)(r+m) L = L+A which reproduces Pi(m+r), together with an integral error formula which involves only the (m + r + I)th derivative of the function approximated. The operator R(m)(m+r) L is defined on functions with r additional orders of smoothness than those on which L is defined. This very general construction holds in all dimensions d. A number of representative examples are considered. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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