Stability of ratio asymptotics of orthogonal polynomials under some class of measure perturbations

Kononova, Anna

doi:10.14232/actasm-013-335-z

Cited by 1 publication

(4 citation statements)

References 6 publications

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“…It is easy to see that the absolute value of this function is constant on each connected component of . Let : As it was shown in [3], this condition is sufficient for the validity of (12).…”

Section: Resultsmentioning

confidence: 81%

“…The main difficulty arising in proving conditions necessary for the behavior of ratio of orthogonal polynomials to be same is the possible cancelation of the zeroes of the functions Θ , , ∈ N (see [3], [11]). In the next lemma we show that if such cancellation arises, such zeroes should belong to .…”

Section: Resultsmentioning

confidence: 99%

“…In work [3] there was found a condition for the measure sufficient (and necessary in some cases) for (1). In the present work, instead of condition (1) we consider a similar condition with a shift: for some fixed ∈ N…”

Section: Introductionmentioning

confidence: 95%

“…The present work continues studying the stability of the asymptotics of the ratio of orthogonal polynomials under a perturbation of the orthogonality measure; the beginning of this study was published in paper [3].…”

Section: Introductionmentioning

confidence: 96%

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On measures generating orthogonal polynomials with similar asymptotic behavior of the ratio at infinity

Kononova¹

2018

Ufimskii Matematicheskii Zhurnal

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We consider the influence of the measure perturbations on the asymptotic behavior of the ratio of orthogonal polynomials. We suppose that the absolutely continuous part of the measure is supported on finitely many Jordan curves. The weight function satisfies the modified Szegö condition. The singular part of the measure consists of finitely many point masses outside the polynomial convex hull of the support of the absolutely continuous part of the measure. We study the stability of asymptotics of the ratio in the following sense: , () , +1 () − , () , +1 () → 0, → ∞. The problem is a generalization of the problem on compactness of the perturbation of Jacobi operator generated by the perturbation of its spectral measure. We find a condition necessary (or necessary and sufficient under some additional restriction) for the stability of the asymptotical behavior of the corresponding orthogonal polynomials. One of the main tools in the study are the Riemann theta functions.

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Section: Resultsmentioning

confidence: 81%

Section: Resultsmentioning

confidence: 99%