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
5 - Lower Bounds for the Number of Nodes
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
Lower bounds for the number of nodes determine the least number of nodes needed for a cubature rule of a given precision. They are essential for studying minimal cubature rules, especially for centrally symmetric integrals. All known lower bounds are collected and discussed in this chapter. These include Möller’s lower bounds for centrally symmetric integrals and wrapped product integrals and several lower bounds for spheres, balls, and simplexes.
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- Minimal Cubature RulesTheory and Practice, pp. 95 - 119Publisher: Cambridge University PressPrint publication year: 2025
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