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
4 - Gauss Cubature Rules
Published online by Cambridge University Press: 10 October 2025
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
Gauss cubature rules are straightforward extensions of the Gauss quadrature rules of one variable. A Gauss cubature rule of degree 2n-1 exists if, and only if, its nodes are common zeros of all orthogonal polynomials of degree n. They are the first example of minimal cubature rules but rarely exist. The chapter gives a comprehensive study that provides a complete characterization of the Gauss cubature rules in terms of the common zeros of the orthogonal polynomials and their structural relations, and it includes examples and counterexamples for the existence of Gauss cubature rules.
Information
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- Minimal Cubature RulesTheory and Practice, pp. 73 - 94Publisher: Cambridge University PressPrint publication year: 2025
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