Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Quadrature Rules and Orthogonal Polynomials
- 2 Cubature Rules: Basics
- 3 Orthogonal Polynomials of Several Variables
- 4 Gauss Cubature Rules
- 5 Lower Bounds for the Number of Nodes
- 6 First Minimal Cubature Rules
- 7 Further Minimal Cubature Rules
- 8 Discrete Fourier Transform and Cubature Rules
- 9 Cubature Rules and Polynomial Ideals
- 10 Epilogue: Two Open Problems
- 11 Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
2 - Cubature Rules: Basics
Published online by Cambridge University Press: aN Invalid Date NaN
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Quadrature Rules and Orthogonal Polynomials
- 2 Cubature Rules: Basics
- 3 Orthogonal Polynomials of Several Variables
- 4 Gauss Cubature Rules
- 5 Lower Bounds for the Number of Nodes
- 6 First Minimal Cubature Rules
- 7 Further Minimal Cubature Rules
- 8 Discrete Fourier Transform and Cubature Rules
- 9 Cubature Rules and Polynomial Ideals
- 10 Epilogue: Two Open Problems
- 11 Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
Summary
This chapter covers the basics of cubature rules. Starting from the definition of polynomial spaces and cubature rules, it discusses interpolatory cubature rules, Tchakaloff’s theorem on positive cubature rules, and Sobolev’s theorem on invariant cubature rules. It provides product-type cubature rules on several regular domains, such as product domains, balls, and simplexes, as well as invariant cubature rules on these domains, and a brief section on constructing cubature rules numerically.
Information
- Type
- Chapter
- Information
- Minimal Cubature RulesTheory and Practice, pp. 19 - 44Publisher: Cambridge University PressPrint publication year: 2025
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