Common zeros of polynomials in several variables and higher dimensional quadrature
Author(s)
Bibliographic Information
Common zeros of polynomials in several variables and higher dimensional quadrature
（Research notes in mathematics, 312）
CRC Press, 2018, c1994
- : hardback
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Note
Bibliography: p. 116-119
Description and Table of Contents
Description
Presents a systematic study of the common zeros of polynomials in several variables which are related to higher dimensional quadrature. The author uses a new approach which is based on the recent development of orthogonal polynomials in several variables and differs significantly from the previous ones based on algebraic ideal theory. Featuring a great deal of new work, new theorems and, in many cases, new proofs, this self-contained work will be of great interest to researchers in numerical analysis, the theory of orthogonal polynomials and related subjects.
Table of Contents
Preface -- 1. Introduction -- 1.1 Review of the theory in one variable -- 1.2 Background to the theory in several variables -- 1.3 Outline of the content -- 2. Preliminaries and Lemmas -- 2.1 Orthogonal polynomials in several variables -- 2.2 Centrally symmetric linear functional -- 2.1 Lemmas -- 3. Motivations -- 3.1 Zeros for a special functional -- 3.2 Necessary conditions for the existence of minimal cubature formula -- 3.3 Definitions -- 4. Common Zeros of Polynomials in Several Variables: First Case -- 4.1 Characterization of zeros -- 4.2 A Christoffel-Darboux formula -- 4.3 Lagrange interpolation -- 4.4 Cubature formula of degree 2n - 1 -- 5. Moller's Lower Bound for Cubature Formula -- 5.1 The first lower bound -- 5.2 Moller's first lower bound -- 5.3 Cubature formulae attaining the lower bound -- 5.4 Moller's second lower bound -- 6. Examples -- 6.1 Preliminaries -- 6.2 Examples: Chebyshev weight function -- 6.3 Examples: product weight function -- 7. Common Zeros of Polynomials in Several Variables: General Case . 85 -- 7.1 Characterization of zeros 86 -- 7.2 Modified Christoffel-Darboux formula 93 -- 7.3 Cubature formula of degree 2n - 1 96 -- 8. Cubature Formulae of Even Degree99 -- 8.1 Preliminaries 99 -- 8.2 Characterization 101 -- 8.3 Example 105 -- 9. Final Discussions 108 -- 9.1 Cubature formula of degree 2n - s 108 -- 9.2 Construction of cubature formula, afterthoughts 112 -- References.
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