Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics
نویسندگان
چکیده
We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L2 convergence of Szegő asymptotics on the spectrum.
منابع مشابه
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Let e ⊂ R be a finite union of disjoint closed intervals. We study measures whose essential support is e and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szeg˝ o condition is equivalent to lim sup a 1 · · · a n cap(e) n > 0 (this includes prior results of Widom and Peherstorfer–Yuditskii). Using Remling's extension of the Denisov–Rakhmanov theorem and an analysis of Jos...
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