Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let μ be a probability measure with an infinite compact support on R . Let us further assume that F n : = f n ∘ ⋯ ∘ f 1 is a sequence of orthogonal polynomials for μ where ( f n ) n = 1 ∞ is a sequence of nonlinear polynomials. We prove that if there is an s 0 ∈ N such that 0 is a root of f n ′ for...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2016-08, Vol.149 (2), p.509-522
Main Author: Alpan, G.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Let μ be a probability measure with an infinite compact support on R . Let us further assume that F n : = f n ∘ ⋯ ∘ f 1 is a sequence of orthogonal polynomials for μ where ( f n ) n = 1 ∞ is a sequence of nonlinear polynomials. We prove that if there is an s 0 ∈ N such that 0 is a root of f n ′ for each n > s 0 then the distance between any two zeros of an orthogonal polynomial for μ of a given degree greater than 1 has a lower bound in terms of the distance between the set of critical points and the set of zeros of some F k . Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-016-0628-8