By Yuan Xu
ISBN-10: 0582246709
ISBN-13: 9780582246706
Offers a scientific examine of the typical zeros of polynomials in different variables that are with regards to greater dimensional quadrature. the writer makes use of a brand new procedure that's in keeping with the hot improvement of orthogonal polynomials in different variables and differs considerably from the former ones in response to algebraic excellent thought. that includes loads of new paintings, new theorems and, in lots of instances, new proofs, this self-contained paintings might be of serious curiosity to researchers in numerical research, the speculation of orthogonal polynomials and comparable matters.
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Extra resources for Common Zeros of Polynominals in Several Variables and Higher Dimensional Quadrature
Example text
On the other hand, if we choose /7 = 0 and y = /, then we obtain a theorem stating necessary and sufficient conditions that Qn+i = Pn+i + FiPn has dimH^ many distinct and real zeros. Actually, this is a statement proved in [40], which leads to a minimal cubature formula of degree 2n, see Section 8. 6. 2. 4 as well as the degree of freedom on the rank of V are both important. 2. A Christoffel-Darboux formula In this section we establish a modified Christoffel-Darboux formula for U'^f„ and Q„+i. 3.
3. Let X i,... 4). Then for 0 < k < n — Uk : ^ ^n+ik "" there exist matrices of full rank, such that Xi , . . ,U ^-lQ 2 n -l. If a Gaussian cubature formula exists, then we have N = dim ll^_i and cro = ... = Let C be a centrally symmetric linear functional defined on H^. 2). This theorem will be used later in Section 5. 2. Necessary conditions for the existence of minimal cubature formula Our second observation is concerned with minimal cubature formulae of degree 2n — 1. 9) such a formula will have N > dim ll^_i many nodes, which follows quite easily from the fact that no polynomial of degree n —1 can vanish on all nodes without violating the square positivity of the linear functional. 1) N = dim ll^_i + a, The restriction that a < 0 < cr < cr G N. Common Zeros of Polynominals in Several Variables and Higher Dimensional Quadrature by Yuan Xu
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