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
DOI:
https://doi.org/10.1090/S0025-5718-00-01206-0
Published electronically:
February 23, 2000
MathSciNet review:
1697644
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Abstract | References | Similar Articles | Additional Information
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 $4m-1$.
- David H. Armitage, Stephen J. Gardiner, Werner Haussmann, and Lothar Rogge, Characterization of best harmonic and superharmonic $L^1$-approximants, J. Reine Angew. Math. 478 (1996), 1–15. MR 1409050, DOI https://doi.org/10.1515/crll.1996.478.1
- Nachman Aronszajn, Thomas M. Creese, and Leonard J. Lipkin, Polyharmonic functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1983. Notes taken by Eberhard Gerlach; Oxford Science Publications. MR 745128
- Dimitar K. Dimitrov, Integration of polyharmonic functions, Math. Comp. 65 (1996), no. 215, 1269–1281. MR 1348043, DOI https://doi.org/10.1090/S0025-5718-96-00747-8
- B. Fuglede, M. Goldstein, W. Haussmann, W. K. Hayman, and L. Rogge (eds.), Approximation by solutions of partial differential equations, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 365, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1168702
- Aldo Ghizzetti and Alessandro Ossicini, Sull’esistenza e unicità delle formule di quadratura gaussiane, Rend. Mat. (6) 8 (1975), 1–15 (Italian, with English summary). MR 380216
- W. K. Hayman and B. Korenblum, Representation and uniqueness theorems for polyharmonic functions, J. Anal. Math. 60 (1993), 113–133. MR 1253232, DOI https://doi.org/10.1007/BF03341969
- Samuel Karlin and Allan Pinkus, An extremal property of multiple Gaussian nodes, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 143–162. MR 0481763
- I. P. Mysovskikh, Interpolyatsionnye kubaturnye formuly, “Nauka”, Moscow, 1981 (Russian). MR 656522
- L. Tschakaloff, General quadrature formulae of Gaussian type, Bulgar. Akad. Nauk. Izv. Mat. Inst. 2(1954), 67–84; English transl., East. J. Approx. 1(1995), 261-276. ;
- P. Turán, On the theory of the mechanical quadrature, Acta Sci. Math. (Szeged) 12(1950), 30–37.
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Additional Information
Borislav D. Bojanov
Affiliation:
Department of Mathematics, University of Sofia, James Boucher 5, 1126 Sofia, Bulgaria
Email:
bor@bgearn.acad.bg
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
Email:
dimitrov@nimitz.dcce.ibilce.unesp.br
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
