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April, 2016 Some remarks on cubature formulas with linear operators
Masatake HIRAO, Takayuki OKUDA, Masanori SAWA
J. Math. Soc. Japan 68(2): 711-735 (April, 2016). DOI: 10.2969/jmsj/06820711


In this paper we consider a novel type of cubature formulas called operator-type cubature formulas. The notion originally goes back to a famous work by G. D. Birkhoff in 1906 on Hermite interpolation problem. A well-known theorem by Sobolev in 1962 on invariant cubature formulas is generalized to operator-type cubature, which provides a systematic treatment of Lebedev's works in the 1970s and some related results by Shamsiev in 2006. We give a lower bound for the number of points needed, and discuss analytic conditions for equality, together with tight illustrations for Laplacian-type cubature.


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Masatake HIRAO. Takayuki OKUDA. Masanori SAWA. "Some remarks on cubature formulas with linear operators." J. Math. Soc. Japan 68 (2) 711 - 735, April, 2016.


Published: April, 2016
First available in Project Euclid: 15 April 2016

zbMATH: 1347.41038
MathSciNet: MR3488142
Digital Object Identifier: 10.2969/jmsj/06820711

Primary: 65D32
Secondary: 05E99 , 15A63

Keywords: cubature formula , Fisher-type inequality , operator-type cubature , Sobolev's theorem , Stroud-type inequality , Sylvester's law of inertia

Rights: Copyright © 2016 Mathematical Society of Japan


Vol.68 • No. 2 • April, 2016
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