Some remarks on cubature formulas with linear operators

• Hirao Masatake

  School of Information and Science Technology, Aichi Prefectural University

• Okuda Takayuki

  Department of Mathematics, Hiroshima University

• Sawa Masanori

  Graduate School of System Informatics, Kobe University

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.

Journal

• Journal of the Mathematical Society of Japan

  Journal of the Mathematical Society of Japan 68 (2), 711-735, 2016

  The Mathematical Society of Japan

Keywords

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

Details

• CRID: 1390282680092151040
• NII Article ID: 130006887144
• NII Article ID: 40020805313
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/06820711
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL BIB ID: 027256957
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I027256957
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed