Regularity of orthogonal rational functions with poles on the unit circle (Q1301981)

Given a sequence \(0=\alpha_0,\alpha_1,\ldots\), with \(|\alpha_k|=1\), \(k>0\), define \(w_0=1\) and \(w_n(z)=(z-\alpha_1)\cdots(z-\alpha_n)\). Consider the space of rational functions generated by the sequence \(f_n=p_n/w_n\) with \(p_n\) a polynomial of degree \(n\), \(n=0,1,\ldots\;\). This paper deals with functions \(\varphi_n\) obtained by orthogonalization of the sequence \(f_0,f_1,\ldots\) with respect to the inner product \(\langle f,g\rangle = \int_{-\pi}^\pi f(e^{i\theta})\overline{g(e^{i\theta})}d\mu(\theta)\) where \(\mu\) is a probability measure on \([-\pi,\pi]\). These orthogonal rational functions have properties similar to properties of polynomials orthogonal on the real line. Among these properties is a generalized 3-term recurrence relation which holds under certain conditions. As derived in [\textit{A. Bultheel, P. González-Vera, E. Hendriksen} and \textit{E. Njåstad}, J. Math. Anal. Appl. 182, No. 1, 221-243 (1994; Zbl 0796.33003)], this recurrence relation, and some other classical properties only hold if the system is regular, meaning that the numerator of \(\varphi_n\) has no zero in \(\alpha_{n-1}\) for all \(n\). In this paper it is investigated what this condition means and whether there do exist regular systems. The latter question is not answered, but it is shown that if there exists a regular system, then any system can be approximated arbitrary well by a sequence of regular systems. A characterization of regularity and properties equivalent to regularity are given. Such properties are, for example, simplicity of the zeros of the \(\varphi_n\), or the fact that these zeros interlace.