18 research outputs found
Filled Julia Sets of Chebyshev Polynomials
We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K⊂ C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K
A question by Chihara about shell polynomials and indeterminate moment problems
The generalized Stieltjes--Wigert polynomials depending on parameters 0\le
p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal
solution concentrated in the zeros of the D-function from the Nevanlinna
parametrization, we obtain a discrete measure \mu^M which is uniquely
determined by its moments. We calculate the coefficients of the corresponding
orthonormal polynomials (P^M_n). As noticed by Chihara, these polynomials are
the shell polynomials corresponding to the maximal parameter sequence for a
certain chain sequence. We also find the minimal parameter sequence, as well as
the parameter sequence corresponding to the generalized Stieltjes--Wigert
polynomials, and compute the value of related continued fractions. The mass
points of \mu^M have been studied in recent papers of Hayman, Ismail--Zhang and
Huber. In the special case of p=q, the maximal parameter sequence is constant
and the determination of \mu^M and (P^M_n) gives an answer to a question posed
by Chihara in 200
Julia Sets of Orthogonal Polynomials
For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K
Szegö's theorem on Parreau-Widom sets
AbstractIn this paper, we generalize Szegőʼs theorem for orthogonal polynomials on the real line to infinite gap sets of Parreau–Widom type. This notion includes Cantor sets of positive measure. The Szegő condition involves the equilibrium measure which in turn is absolutely continuous. Our approach builds on a canonical factorization of the M-function and the covering space formalism of Sodin–Yuditskii
The moment problem associated with the Stieltjes-Wigert polynomials
AbstractWe consider the indeterminate Stieltjes moment problem associated with the Stieltjes–Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes–Wigert polynomials, expressions for the entire functions A, B, C, and D from the Nevanlinna parametrization are obtained. We describe T(n)(μ) for n∈N when μ=μ0 is a particular N-extremal solution and explain in detail what happens when n→∞
Finite and Infinite Gap Jacobi Matrices
The present paper reviews the theory of bounded Jacobi matrices whose essential spectrum is a finite gap set, and it explains how the theory can be extended to also cover a large number of infinite gap sets. Two of the central results are generalizations of Denisov–Rakhmanov’s theorem and Szegő’s theorem, including asymptotics of the associated orthogonal polynomials. When the essential spectrum is an interval, the natural limiting object J0 has constant Jacobi parameters. As soon as gaps occur, ℓ say, the complexity increases and the role of J0 is taken over by an ℓ -dimensional isospectral torus of periodic or almost periodic Jacobi matrices