## OP-SF WEB

### Extract from OP-SF NET

Topic #10 -------------- OP-SF NET --------------- February 9, 1995
~~~~~~~~~
From: Yuan Xu
Subject: New book on Orthogonal Polynomials in Several Variables
The following Research Note appeared:
Yuan Xu
"Common Zeros of Polynomials in Several Variables and Higher
Dimensional Quadrature"
Pitman Research Notes in Mathematics Series, 312
Essex, 1994.
Abstract:
This research note presents a systematic study of the common zeros of
sets of polynomials in several variables which are related to higher
dimensional quadrature (often called cubature). Just like the classical
Gaussian quadrature formula, a cubature formula of degree 2n-1 needs at
least N nodes, where N denotes the dimension of the subspace of
polynomials of degree at most n-1. A cubature formula whose number of
nodes is equal to N exists if, and only if, the corresponding orthogonal
polynomials of degree n have N real and distinct common zeros. However,
since such a cubature does not exist in general, one is led to study
cubature formulae with more nodes, the most interesting cases being the
ones with a minimal number of nodes.
In the recent development of orthogonal polynomials in several
variables, the common zeros of orthogonal polynomials are characterized
as the joint eigenvalues of a family of block Jacobi matrices defined
through the coefficient matrices of a three-term relation in
vector-matrix form. The main results in this book show that, in general,
cubature formulae are based on the real and distinct common zeros of a
family of quasi-orthogonal polynomials, and the existence of these common
zeros can be characterized through the solvability of certain nonlinear
matrix equations derived from the modified block Jacobi matrices.
The approach is parallel to that of one variable, which differs
significantly from earlier ones based on the algebraic ideal theory.
The book is in essence a research paper; a good portion of the
theorems are new and, in many cases, new proofs are given for known
results. The content is basically self-contained, including one
section that summarizes the recent results on the general structure of
orthogonal polynomials in several variables.

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