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Majorization Results for Zeros of Orthogonal Polynomials

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Majorization results for zeros of orthogonal polynomials ∗ Walter Van Assche KU Leuven September 4, 2018 Abstract We show that the zeros of consecutive orthogonal polynomials pn and pn−1 are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of pn (1) and the associated polynomial pn−1 and for the zeros of the polynomial obtained by deleting the kth row and column (1 ≤ k ≤ n) in the corresponding Jacobi matrix. 1 Introduction 1.1 Majorization Recently S.M. Malamud [3, 4] and R. Pereira [6] have given a nice extension of the Gauss- Lucas theorem that the zeros of the derivative p′ of a polynomial p lie in the convex hull of the roots of p. They both used the theory of majorization of sequences to show that if x =(x1, x2,...,xn) are the zeros of a polynomial p of degree n, and y =(y1,...,yn−1) are the zeros of its derivative p′, then there exists a doubly stochastic (n − 1) × n matrix S such that y = Sx ([4, Thm. 4.7], [6, Thm. 5.4]). A rectangular (n − 1) × n matrix S =(si,j) is said to be doubly stochastic if si,j ≥ 0 and arXiv:1607.03251v1 [math.CA] 12 Jul 2016 n n− 1 n − 1 s =1, s = . i,j i,j n j=1 i=1 X X In this paper we will use the notion of majorization to rephrase some known results for the zeros of orthogonal polynomials on the real line. It is an open problem to see what can be said about the zeros of orthogonal polynomials in the complex plane in terms of majorization. Let us begin by defining the notion of majorization. An excellent source of information on majorization is the book by Marshall and Olkin (and Arnold, who was added for the second edition) [5]. See also Horn and Johnson [2, Def. 4.3.24]. ∗This research was supported by KU Leuven research grant OT/12/073 and FWO research project G.0864.16N. 1 n n Definition 1.1. let x = (x1, x2,...,xn) ∈ R and y = (y1,y2,...,yn) ∈ R . Then the vector y is said to majorize the vector x, or x is majorized by y (notation: x ≺ y) if xˆ1 + ··· +ˆxj ≤ yˆ1 + ··· +ˆyj, 1 ≤ j ≤ n − 1, and xˆ1 + ··· +ˆxn =y ˆ1 + ··· +ˆyn, where xˆ = (ˆx1,..., xˆn) and yˆ = (ˆy1,..., yˆn) contain the components of x and y in decreas- ing order. The following result (of Hardy, Littlewood and P´olya) gives an alternative way to define majorization in terms of doubly stochastic matrices. Recall that a square matrix n A =(ai,j)i,j=1 of order n is doubly stochastic if ai,j ≥ 0 and n n ai,j =1, ai,j =1, i=1 j=1 X X i.e., the elements in every row and every column add up to 1. Theorem 1.1. Let x =(x1,...,xn) and y =(y1,...,yn). Then x ≺ y if and only if there exists a doubly stochastic matrix A such that x = Ay. See [5, Thm. 2.B.2], [2, Thm. 4.3.33]. Note that the doubly stochastic matrix A need not be unique. 1.2 Orthogonal polynomials Let µ be a positive measure on the real line for which all the moments exist. To simplify notation, we will assume that µ is a probability measure so that µ(R) = 1. The orthogonal polynomials pn for this measure satisfy the orthogonality relations pn(x)pm(x) dµ(x)= δm,n, Z n and we denote the orthonormal polynomials by pn(x) = γnx + ··· , with the conven- tion that γn > 0. Orthogonal polynomials on the real line always satisfy a three term recurrence relation xpn(x)= an+1pn+1(x)+ bnpn(x)+ anpn−1(x), (1.1) with an > 0 for n ≥ 1 and bn ∈ R for n ≥ 0. These recurrence coefficients are given by γ a = xp (x)p (x) dµ(x)= n−1 , (1.2) n n n−1 γ Z n 2 bn = xpn(x) dµ(x). (1.3) Z The zeros of pn are all real and simple and we will denote them by x1,n < x2,n < ··· < xn,n, hence in increasing order. It is well known that the zeros of pn−1 and pn interlace, i.e., xj,n < xj,n−1 < xj+1,n, 1 ≤ j ≤ n − 1. 2 The zeros of pn are also the eigenvalues of the tridiagonal matrix (Jacobi matrix) b0 a1 0 0 ··· 0 a b a 0 ··· 0 1 1 1 0 a2 b2 a3 0 Jn = . (1.4) . .. .. .. 0 ··· 0 an−2 bn−2 an−1 0 ··· 0 0 an−1 bn−1 and the interlacing of the zeros of pn and pn−1 hence corresponds to the interlacing of the eigenvalues of Jn and Jn−1. Another consequence is that for the trace of Jn one has Tr Jn = x1,n + x2,n + ··· xn,n = b0 + b1 + ··· + bn−1, which gives a relation between the sum of the zeros of pn and a partial sum of the recurrence coefficients (bn)n≥0. In particular we find x1,n−1 + x2,n−1 + ··· xn−1,n−1 + bn−1 = x1,n + x2,n + ··· + xn,n. (1.5) The interlacing of the zeros of pn−1 and pn, together with (1.5) then imply that the vector (x1,n−1,...,xn−1,n−1, bn−1) is majorized by (x1,n, x2,n,...,xn,n). In Section 2 we will give an explicit expression for a doubly stochastic matrix A for which x1,n−1 x1,n x2,n−1 x2,n . . . = A . xn−1,n−1 xn−1,n bn−1 xn,n Another interesting interlacing property involves the zeros of the associated polynomial p (z) − p (x) p(1) (z)= a n n dµ(x), n−1 1 z − x Z see, e.g., [8]. This polynomial appears naturally as the numerator of the Pad´eapproximant to the function ∞ dµ(x) m f(z)= a = a k , m = xk dµ(x), 1 z − x 1 zk+1 k k Z X=0 Z near infinity, i.e., (1) n+1 pn(z)f(z) − pn−1(z)= O(1/z ), z →∞. (1) The zeros of pn−1 are again real and simple and we denote them by y1,n−1 < y2,n−1 < ··· <yn−1,n−1. They interlace with the zeros of pn in the sense that xj,n <yj,n−1 < xj+1,n, 1 ≤ j ≤ n − 1. 3 In fact, the partial fractions decomposition of the Pad´eapproximant is (1) n pn−1(z) λk,n = a1 , (1.6) pn(z) z − xk,n k X=1 where the λk,n are the Christoffel numbers 1 1 λk,n = = . (1.7) n−1 2 ′ j=0 pj (xk,n) anpn(xk,n)pn−1(xk,n) (1) P (1) The zeros of pn−1 are equal to the eigenvalues of the Jacobi matrix Jn obtained from Jn by deleting the first row and first column: b1 a2 0 0 ··· 0 a b a 0 ··· 0 2 2 3 (1) 0 a3 b3 a4 0 Jn = . . (1.8) . .. .. .. 0 ··· 0 an−2 bn−2 an−1 0 ··· 0 0 an−1 bn−1 For the trace we therefore have (1) Tr Jn = y1,n−1 + y2,n−1 + ··· + yn−1,n−1 = b1 + b2 + ··· + bn−1, and hence y1,n−1 + y2,n−1 + ··· + yn−1,n−1 + b0 = x1,n + x2,n + ··· + xn,n. (1.9) Combining the interlacing property and (1.9) we then see that (y1,n−1,...,yn−1,n−1, b0) is majorized by (x1,n, x2,n,...,xn,n). In Section 3 we will give an explicit expression for a doubly stochastic matrix B for which y1,n−1 x1,n y2,n−1 x2,n . . . = B . yn−1,n−1 xn−1,n b0 xn,n Finally, in Section 4 we will prove a similar result for the zeros of the characteristic polynomial of the matrix obtained by deleting the kth row and column of the Jacobi matrix Jn. We will frequently use Gaussian quadrature using the zeros of orthogonal polynomials. Suppose that supp(µ) ⊂ [a, b] and that {xj,n, 1 ≤ j ≤ n} are the zeros of the orthogonal polynomial pn and {λj,n, 1 ≤ j ≤ n} the corresponding Christoffel numbers, then for every function f ∈ C2n([a, b]) n f (2n)(ξ) f(x) dµ(x)= λ p(x )+ , (1.10) j,n j,n (2n)!γ2 j=1 n Z X 4 where γn is the leading coefficient of pn and ξ ∈ (a, b) (see, e.g., [1, §1.4.2], [7, §3.4]). The quadrature sum therefore gives the integral exactly whenever f is a polynomial of degree ≤ 2n − 1. The Christoffel numbers λj,n given in (1.7) are also given by b p2 (x) λ = n dµ(x) > 0. (1.11) j,n (x − x )2[p′ (x ]2 Za j,n n j,n 2 Zeros of consecutive orthogonal polynomials Theorem 2.1. Let (x1,n−1,...,xn−1,n−1) be the zeros of pn−1 and (x1,n,...,xn,n) be the zeros of pn, and let (an+1, bn)n≥0 be the recurrence coefficients in (1.1) and (1.4). Then x1,n−1 x1,n x2,n−1 x2,n . . . = A . , (2.1) xn−1,n−1 xn−1,n bn−1 xn,n n where A =(ai,j)i,j=1 is a doubly stochastic matrix with entries 2 2 2 a λj,np (xj,n)λk,n p (xi,n ) n n−1 −1 n −1 , 1 ≤ i ≤ n − 1, (x − x )2 ai,j = j,n i,n−1 (2.2) 2 λj,npn−1(xj,n), i = n.
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