Browsing by Subject "Parreau-Widom sets"

Now showing 1 - 4 of 4

Open Access
Asymptotic properties of Jacobi matrices for a family of fractal measures
(Taylor and Francis, 2018) Alpan, G.; Goncharov, A.; Şimşek, A. N.
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined, and different aspects of orthogonal polynomials on them were studied recently. Our main aim is to numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on (Formula presented.).
Open Access
Asymptotics of extremal polynomials for some special cases
(2017-05) Alpan, Gökalp
We study the asymptotics of orthogonal and Chebyshev polynomials on fractals. We consider generalized Julia sets in the sense of Br uck-B uger and weakly equilibrium Cantor sets which was introduced in [62]. We give characterizations for Parreau-Widom condition and optimal smoothness of the Green function for the weakly equilibrium Cantor sets. We also show that, for small parameters, the corresponding Hausdor measure and the equilibrium measure of a set from this family are mutually absolutely continuous. We prove that the sequence of Widom-Hilbert factors for the equilibrium measure of a non-polar compact subset of R is bounded below by 1. We give a su cient condition for this sequence to be unbounded above. We suggest de nitions for the Szeg}o class and the isospectral torus for a generic subset of R
Open Access
Numerical study of orthogonal polynomials for fractal measures
(2016-07) Şimşek, Ahmet Nihat
In recent years, potential theory has an essential effect on approximation theory and orthogonal polynomials. Basic concepts of the modern theory of general orthogonal polynomials are described in terms of Potential Theory. One of these concepts is the Widom factors which are the ratios of norms of extremal polynomials to a certain degree of capacity of a set. While there is a theory of Widom factors for finite gap case, very little is known for fractal sets, particularly for supports of continuous singular measures. The motivation of our numerical experiments is to get some ideas about how Widom factors behave on Cantor type sets. We consider weakly equilibrium Cantor sets, introduced by A.P. Goncharov in [16], which are constructed by iteration of quadratic polynomials that change from step to step depending on a sequence of parameters. Changes in these parameters provide a Cantor set with several desired properties. We give an algorithm to calculate recurrence coefficients of orthogonal polynomials for the equilibrium measure of such sets. Our numerical experiments point out stability of this algorithm. Asymptotic behaviour of the recurrence coefficients and the zeros of orthogonal polynomials for the equilibrium measure of four model Cantor sets are studied via this algorithm. Then, several conjectures about asymptotic behaviour of the recurrence coefficients, Widom factors, and zero spacings are proposed based on these numerical experiments. These results are accepted for publication [1] (jointly with G. Alpan and A.P. Goncharov).
Open Access
Orthogonal Polynomials on Generalized Julia Sets
(Birkhauser Verlag AG, 2017) Alpan, G.; Goncharov, A.
We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green’s functions and Parreau–Widom criterion for a special family of real generalized Julia sets. © 2017, Springer International Publishing.