2010/01/04 by Franz Peherstorfer, Peherstorfer, Franz · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.CA #math.CV
paper · pdf · doi:10.48550/arxiv.1001.0478
The last modifications and corrections of this manuscript were done by the author in the two months preceding this passing away in November 2009. The manuscript is not published elsewhere
arxiv created 2010/01/04 · openalex publication_date 2010/01/04 · arxiv updated 2010/01/14 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Let E = ∪j = 1l [a2j-1,a2j], a1 < a2 < ... < a2l, l ≥ 2 and set \boldmathω(∞) =(ω1(∞),...,ωl-1(∞)), where ωj(∞) is the harmonic measure of [a2 j - 1, a2 j] at infinity. Let μ be a measure which is on E absolutely continuous and satisfies Szegő's-condition and has at most a finite number of point measures outside E, and denote by (Pn) and (\mathcal Qn) the orthonormal polynomials and their associated Weyl solutions with respect to dμ, satisfying the recurrence relation √λ2 + n y1 + n = (x - α1 + n) yn -√λ1 + n y-1 + n. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence (n \boldmathω(∞))n∈ \mathbb N modulo 1; More precisely, putting (\boldmathαl-11 + n, \boldmathλl-12 + n) = (α[(l 1)/(2)]+1+n,..., α1+n,..., α-[(l-2)/(2)]+1+n, λ[(l-2)/(2)]+2+n, ...,λ2+n, ..., λ-[(l-1)/(2)]+2+n) we prove that (\boldmathαl-11 + nν, \boldmathλl-12 + nν)ν∈ \mathbb N converges if and only if (nν\boldmathω(∞))ν∈ \mathbb N converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of (\boldmathαl-11 + n, \boldmathλl-12 + n) and (n\boldmathω(∞)) modulo 1.