By Yuan Xu
ISBN-10: 0582246709
ISBN-13: 9780582246706
Provides a scientific learn of the typical zeros of polynomials in numerous variables that are relating to better dimensional quadrature. the writer makes use of a brand new procedure that's in accordance with the new improvement of orthogonal polynomials in numerous 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 situations, new proofs, this self-contained paintings may be of significant 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
Sample text
Since all eigenvalues of symmetric matrices are real-valued, we conclude that all zeros of Q are real. Moreover, it is well-known that symmetric matrices are diagonalizable, and a family of diagonalizable matrices is simultaneously diagonalizable if, and only if, it is commuting [9, p. 52]. To show that 5n 1, . . 13). 6). 5), holds true. 7). 7) imply that 5 n ,i,.. • Sn,d are symmetric and commuting, thus, simultaneously diagonalizable. Hence, 5 n ,i,. j + a many real joint eigenvalues which are common zeros of f7^Pn and Qn+i- That these zeros are pairwise distinct follows from the fact that at each zero at least one of the partial derivatives of ¡7 ^P n and Q n+i does not vanish.
11), the first term can be written as (y^ —a;,)Kn(x,y). 2). 2) and then let y,- —> x,-. I If A is a zero of Q = {i7^P„, Qn+i} and at least one partial derivative of U'^Fn or Qn+i at A is unequal to zero, then we say that A is a simple zero of Q. 2. 2,1. (Q)„+i (x ) + 44 diU'^Fnix)]. Moreover, all zeros of Q are simple. Proof. 1 by taking X = Xjt and y = Xj. By the definition of •) it is evident that Kn^^(x,x) > 0 for every x, in particular for Xjt. 7) implies that 5,(i7^Pn) and ^¿Qn+i cannot be both zero.
Tn —A:H-2,Ar—1 If Jjl\ x ) = (1 —x^)^ and I'^x) = (1 — ek,n = 0. a > —1, then Pj^ are the classical Jacobi polynomials on the unit disk D which are orthogonal with respect to the weight function (1 — —y^)". 3. Let // be a nondecreasing function on R with finite moments. Let {Pn}^=o be orthonormal polynomials with respect to ¡i\ it satisfies the three-term relation ( 1. 1. 1). 16) Then X— are polynomials of total degree n in the variables u and v. Moreover, let W{u^v) = /i'(x)//'(y); it is not hard to verify that are orthonormal polynomials in two variables with respect to the measures da = (u^ - 4v)'^^^‘^ W{u,v)dudv, on i?
Common Zeros of Polynominals in Several Variables and Higher Dimensional Quadrature by Yuan Xu
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