An Algorithmic Implementation of the Generalized Christoffel Theorem

Abstract

Let dλ(t) be a nonnegative measure on some interval I ⊂ ℝ, with λ(t) having infinitely many points of increase, and assume that all moments ${\mu _r} = \int {_I{t^t}d\lambda (t)}$ exist, with μ₀ > 0. Let π_n(t) = π_n (t;dλ), n = 0, 1, 2,..., be the associated system of (monic) orthogonal polynomials. Given two polynomials $u(t) = \pm \mathop \prod \limits_{\lambda = 1}^\ell (t - {u_\lambda }),v(t) = \mathop \prod \limits_{\mu = 1}^m (t - {v_\mu })$, with pairwise distinct roots, and such that [u(t)/v(t)]dλ(t) is nonnegative on I and has finite moments of all orders, the generalized Christoffel theorem expresses the orthogonal polynomials relative to the measure [u(t)/v(t)]dλ(t) in determinantal form in terms of the polynomials {π_n}. Assuming, for example, that m ≤ n, one has

$$u(t){\Pi _n}(t;\frac{u}{v}d\lambda ) = const. \times $$

where

$$\left| \begin{gathered}{\pi _{n - m}}(t) \cdots {\pi _{n - 1}}(t){\pi _n}(t){\pi _{n + 1}}(t) \cdots {\pi _{n + \ell }}(t) \hfill \\{\pi _{n - m}}({u_1}) \cdots {\pi _{n - 1}}({u_1}){\pi _n}({u_1}){\pi _{n + 1}}({u_1}) \cdots {\pi _{n + \ell }}({u_1}) \hfill \\\ldots .................... \hfill \\{\pi _{n - m}}({u_\ell }) \cdots {\pi _{n - 1}}({u_\ell }){\pi _n}({u_\ell }){\pi _{n + 1}}({u_\ell }) \cdots {\pi _{n + \ell }}({u_\ell }) \hfill \\{\rho _{n - m}}({v_1}) \cdots {\rho _{n - 1}}({v_1}){\rho _n}({v_1}){\rho _{n + 1}}({v_1}) \cdots {\rho _{n + \ell }}({v_1}) \hfill \\\ldots .................... \hfill \\{\rho _{n - m}}({v_m}) \cdots {\rho _{n - 1}}({v_m}){\rho _n}({v_m}){\rho _{n + 1}}({v_m}) \cdots {\rho _{n + \ell }}({v_m}) \hfill \\\end{gathered} \right|$$

((1.1))

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Sponsored in part by the National Science Foundation under grant MCS-7927158.