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

Springer Science and Business Media LLC - 2023
Maŕıa José Cantero1, Leandro del Moral Ituarte1, L. Velázquez1
1Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza, Zaragoza, Spain

Tóm tắt

AbstractThe 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 $$p_n^2\mathrm{d}\mu $$ p n 2 d μ , where $$p_n$$ p n are the orthonormal polynomials with respect to a measure $$\mu $$ μ 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 $$\mu $$ μ supported on an unbounded subset of the real line yields a weakly convergent sequence $$p_n^2\mathrm{d}\mu $$ p n 2 d μ (Corollary 8.10), but there exist instances such that $$p_n^2\mathrm{d}\mu $$ p n 2 d μ becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of $$p_n^2\mathrm{d}\mu $$ p n 2 d μ 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.

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