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Mathematics of Computation

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Integration of polyharmonic functions

Author: Dimitar K. Dimitrov
Journal: Math. Comp. 65 (1996), 1269-1281
MSC (1991): Primary 31B30; Secondary 65D32
MathSciNet review: 1348043
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Abstract: The results in this paper are motivated by two analogies. First, $m$-harmonic functions in $\mathbf {R}^{n}$ are extensions of the univariate algebraic polynomials of odd degree $2m-1$. Second, Gauss’ and Pizzetti’s mean value formulae are natural multivariate analogues of the rectangular and Taylor’s quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules could be generalized to results about integration of polyharmonic functions. This is done for the Tchakaloff-Obrechkoff quadrature formula and for the Gaussian quadrature with two nodes.

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Additional Information

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, polyharmonic monospline
Received by editor(s): August 15, 1994
Received by editor(s) in revised form: May 22, 1995
Additional Notes: Research supported by the Bulgarian Ministry of Science under Grant MM-414 and The Royal Society London
Article copyright: © Copyright 1996 American Mathematical Society