Wall Polynomials on the Real Line: A Classical Approach to OPRL Khrushchev’s Formula

Abstract

The standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (Grünbaum and Velázquez in Adv Math 326:352–464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchev’s formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchev’s formula is used to rederive the characterization given in Simon (J Approx Theory 126:198–217, 2004) for the weak convergence of p_n^2mathrm{d}mu , where p_n are the orthonormal polynomials with respect to a measure mu supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchev’s formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure mu supported on an unbounded subset of the real line yields a weakly convergent sequence p_n^2mathrm{d}mu (Corollary 8.10), but there exist instances such that p_n^2mathrm{d}mu becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of p_n^2mathrm{d}mu in the unbounded case are obtained via Khrushchev’s formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchev’s formula on the real line which sheds light on its graph theoretical meaning, linked to Pólya’s recurrence theory for classical random walks.