A new approach to ratio asymptotics for orthogonal polynomials

@inproceedings{Simanek2011ANA,
  title={A new approach to ratio asymptotics for orthogonal polynomials},
  author={Brian Simanek},
  year={2011}
}
We use a non-linear characterization of orthonormal polynomials due to Saff in order to prove an equivalence of norm asymptotics and strong asymptotics for polynomials. Several applications of this equivalence are also discussed. One of our main results is that for regular measures on the closed unit disk - including, but not limited to the unit circle - or the interval [-2,2], one has ratio asymptotics along a sequence of asymptotic density 1. 
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References

SHOWING 1-10 OF 35 REFERENCES
REMARKS ON RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS
Using a nonlinear integral characterization of orthogonal polynomials in the complex plane, we provide a simple method for deducing a weak form of relative asymptotics exterior to the convex hull of…
Ratio and Relative Asymptotics of Polynomials Orthogonal on an Arc of the Unit Circle
Ratio and relative asymptotics are given for sequences of polynomials orthogonal with respect to measures supported on an arc of the unit circle, where their absolutely continuous component is…
Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line
  • B. Simon
  • Mathematics
    J. Approx. Theory
  • 2004
Ratio asymptotics and quadrature formulas
A technique to find the asymptotic behavior of the ratio between a polynomialssn and thenth orthonormal polynomial with respect to a positive measureμ is shown. Using it, some new results are found…
Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures
Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions
Asymptotics of orthogonal polynomials and point perturbation on the unit circle
General Orthogonal Polynomials
Introduction 1. Upper and lower bounds 2. Zero distribution of orthogonal polynomials 3. Regular n-th root asymptotic behaviour of orthogonal polynomials 4. Regularity criteria 5. Localization 6.…
Orthogonal polynomials with ratio asymptotics
A general construction is given for measures for which the corresponding orthogonal polynomials have ratio asymptotics. Let v be a positive Borel measure on the unit circle T and form the orthonormal…
On the Asymptotics of the Ratio of Orthogonal Polynomials. Ii
Let be a positive measure on the circumference and let almost everywhere on . Let be the orthogonal polynomials corresponding to , and let be their parameters. Then .Bibliography: 5 titles.
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