Orthogonal polynomials on arcs of the unit circle. I
In this paper polynomials \(P_n\) orthogonal on the unit circle with respect to a weight function of the form \(f(\theta)= {\mathcal W}(\theta)/{\mathcal A}(\theta) r(\theta)\) for \(\theta\in E_\ell= \bigcup^\ell_{j=1} [\theta_{2j-1}, \theta_{2j}]\) are investigated. Here \({\mathcal A}\) is a real trigonometric polynomial which has no zeros on \(E_\ell\), \({\mathcal W}\) is a trigonometric polynomial vanishing at some of the endpoints \(\theta_k\) and \(r(\theta)= (-1)^j \sqrt{|{\mathcal R}(\theta)|}\) when \(\theta\in (\theta_{2j-1}, \theta_{2j})\), where \({\mathcal R}\) is the trigonometric polynomial vanishing at all the endpoints \(\theta_k\). Observe that the weight function \(f\) is not necessarily positive on \(E_\ell\). First, a few examples are given (for two intervals), which are useful in understanding the kind of problems considered in this paper. It is shown that \(P_n\) is orthogonal with respect to \(f\) if and only if there exists a polynomial \(Q_m\) and a polynomial \(g_n\) such that \(W(z) P^2_n(z)- V(z) Q^2_m(z)= z^n A(z)g_n(z)\). Furthermore, the polynomial \(Q_m\) can then be represented in terms of the orthogonal polynomials \(P_n\) and the polynomials of the second kind. These orthogonal polynomials are interesting because they are useful explicit examples of orthogonal polynomials on several arcs, in particular polynomials with periodic reflection coefficients (from a certain index onward). The paper is rather technical but deserves close attention.
scientific article; zbMATH DE number 885477