Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1Quadrature Rules and Orthogonal Polynomials
- 2Cubature Rules: Basics
- 3Orthogonal Polynomials of Several Variables
- 4Gauss Cubature Rules
- 5Lower Bounds for the Number of Nodes
- 6First Minimal Cubature Rules
- 7Further Minimal Cubature Rules
- 8Discrete Fourier Transform and Cubature Rules
- 9Cubature Rules and Polynomial Ideals
- 10Epilogue: Two Open Problems
- 11Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
6 - First Minimal Cubature Rules
Published online by Cambridge University Press: 10 October 2025
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1Quadrature Rules and Orthogonal Polynomials
- 2Cubature Rules: Basics
- 3Orthogonal Polynomials of Several Variables
- 4Gauss Cubature Rules
- 5Lower Bounds for the Number of Nodes
- 6First Minimal Cubature Rules
- 7Further Minimal Cubature Rules
- 8Discrete Fourier Transform and Cubature Rules
- 9Cubature Rules and Polynomial Ideals
- 10Epilogue: Two Open Problems
- 11Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
Summary
The first minimal cubature rules beyond the Gauss cubature rules of degree 2n-1 are generated by zeros of orthogonal polynomials of degree n and quasi-orthogonal polynomials of degree n+1, which include those that attain Möller’s lower bound. This chapter provides a complete characterization of such cubature rules. The characterization is given in terms of a family of nonlinear equations derived through the structural relations of orthogonal polynomials. Several families of minimal or near-minimal cubature rules for the Chebyshev weight functions on the square are presented.
Information
- Type
- Chapter
- Information
- Minimal Cubature RulesTheory and Practice, pp. 120 - 157Publisher: Cambridge University PressPrint publication year: 2025
Accessibility standard: WCAG 2.2 A
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
The PDF of this chapter complies with version 2.2 of the
Web Content Accessibility Guidelines
(WCAG), offering more comprehensive accessibility measures for a broad range
of users and meets the basic
(A)
level of WCAG compliance, addressing essential accessibility barriers.
Content Navigation
Table of contents navigation
Allows you to navigate directly to chapters, sections, or non‐text items through a linked table of contents, reducing the need for extensive scrolling.
Allows you to navigate directly to chapters, sections, or non‐text items through a linked table of contents, reducing the need for extensive scrolling.
Index navigation
Provides an interactive index, letting you go straight to where a term or subject appears in the text without manual searching.
Provides an interactive index, letting you go straight to where a term or subject appears in the text without manual searching.
Reading Order & Textual Equivalents
Single logical reading order
You will encounter all content (including footnotes, captions, etc.) in a clear, sequential flow, making it easier to follow with assistive tools like screen readers.
You will encounter all content (including footnotes, captions, etc.) in a clear, sequential flow, making it easier to follow with assistive tools like screen readers.
Short alternative textual descriptions
You get concise descriptions (for images, charts, or media clips), ensuring you do not miss crucial information when visual or audio elements are not accessible.
You get concise descriptions (for images, charts, or media clips), ensuring you do not miss crucial information when visual or audio elements are not accessible.