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Gaussian extended cubature formulae for polyharmonic functions

Authors: Borislav D. Bojanov and Dimitar K. Dimitrov
Journal: Math. Comp. 70 (2001), 671-683
MSC (2000): Primary 31B30; Secondary 65D32
Published electronically: February 23, 2000
MathSciNet review: 1697644
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Abstract: The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses $m$ values of linear functionals (integrals over hyperspheres) and is exact for all $2m$-harmonic functions, and consequently, for all algebraic polynomials of $n$ variables of degree $4m-1$.

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Additional Information

Borislav D. Bojanov
Affiliation: Department of Mathematics, University of Sofia, James Boucher 5, 1126 Sofia, Bulgaria

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
MR Author ID: 308699

Keywords: Polyharmonic function, extended cubature formula, polyharmonic order of precision, Gaussian extended cubature formula
Received by editor(s): January 6, 1998
Received by editor(s) in revised form: May 5, 1999
Published electronically: February 23, 2000
Additional Notes: The research was done during the visit of the first author to UNESP, São José do Rio Preto, which was supported by the Brazilian Foundation FAPESP under Grant 96/07748-3.
The research of the second author is supported by the Brazilian foundations CNPq under Grant 300645/95-3 and FAPESP under Grant 97/06280-0. The research of both authors is supported by the Bulgarian Science Foundation under Grant MM-802/98.
Article copyright: © Copyright 2000 American Mathematical Society