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

Abstract

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n{)}_{n=1}^{\infty }$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n{)}_{n=1}^{\infty }$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ \cdots \circ f_1$ for all $n\in \mathbb{N}$. We prove that if there is an $s_0\in \mathbb{N}$ such that $0$ is a root of $f_n^{\prime }$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $\mu$ 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.

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References in corpus (4)

• Reflectionless measures with a point mass and singular continuous component
• Orthogonal polynomials for the weakly equilibrium Cantor sets
• Two Measures on Cantor Sets
• Chebyshev polynomials on generalized Julia sets

Cited by in corpus (1)

• Orthogonal polynomials on Cantor sets of zero Lebesgue measure