On Bernstein-Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients

AbstractGeronimus has shown that a sequence of orthogonal polynomials (pn) with periodic recurrence coefficients for n ⩾ n0 is orthogonal on a set of disjoint intervals el = Uj = 1l = [a2j − 1, a2j] with respect to a distribution of the form dψ(x)=−∏j=12l(x−aj|pv(x)|dx+dμ(x) where ρv,(x) = Πj = 1v (x − Wj) with sgn νv(x) = (−1)l + 1 − j on (a2j − 1, a2j) for j = 1, …, l, v ⩾ l − 1, and where μ is a certain point measure with supp(μ) ⊂{w1, …, wv}. In this paper we show (in fact a more general result is presented) that a sequence of polynomials (pn) orthogonal with respect to dψ has recurrence coefficients of period N, N ⩾ l, for n ⩾ n0, if and only if there exists a so-called Chebyshev polynomial TN of degree N on El, where a polynomial TN is called a Chebyshev polynomial on El if ¦TN| attains its maximum value on El at N + l points from El. Furthermore it is demonstrated how to get in a simple way a (nonlinear) recurrence relation for the recurrence coefficients of the orthogonal polynomials. Results on Chebyshev polynomials on several intervals are also given