Elliptic Orthogonal and Extremal Polynomials

Abstract

In this paper we give, in terms of elliptic functions, an explicit representation of the polynomials which are orthogonal on E = [−1, α] ∪ [β, 1] with respect to a weight functions of the form √{(1 − x2)(x − α)(x − β)}/∣ρ(x)∣ (in fact a more general class of weight functions, not necessarily positive on E, is considered), where ρ is an arbitrary polynomial which has no zero in int(E). Also closed expressions are provided for the recurrence coefficients of the orthogonal polynomials. Furthermore it is shown that under certain special conditions these orthogonal polynomials are also miniamal polynomials on E with respect to the maximum norm with weight 1/√∣ρ(x)∣. Then, for a large class of weight functions (including the above-mentioned ones), the support of which consists of two intervals which have non-rational harmonic measure at ∞, it is shown that the recurrence coefficients of the orthogonal polynomials have an infinite set of limit points. This fact is rather surprising, because in the single-interval case it is well known that the recurrence coefficients of the polynomials orthogonal with respect to a weight function positive almost everywhere on the interval converge. Finally we give the explicit solution of periodic and non-periodic Toda lattices the band structure of which consists of two intervals.