Journal of the Mathematical Society of Japan

Some remarks on cubature formulas with linear operators

Masatake HIRAO, Takayuki OKUDA, and Masanori SAWA

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Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 2 (2016), 711-735.

Dates
First available in Project Euclid: 15 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1460727377

Digital Object Identifier
doi:10.2969/jmsj/06820711

Mathematical Reviews number (MathSciNet)
MR3488142

Zentralblatt MATH identifier
1347.41038

Subjects
Primary: 65D32: Quadrature and cubature formulas
Secondary: 05E99: None of the above, but in this section 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx]

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

Citation

HIRAO, Masatake; OKUDA, Takayuki; SAWA, Masanori. Some remarks on cubature formulas with linear operators. J. Math. Soc. Japan 68 (2016), no. 2, 711--735. doi:10.2969/jmsj/06820711. https://projecteuclid.org/euclid.jmsj/1460727377


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