Browsing by Subject "Widom factors"
Now showing 1 - 4 of 4
- Results Per Page
- Sort Options
Item 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.).Item Open Access Asymptotics of extremal polynomials for some special cases(2017-05) Alpan, GökalpWe 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 RItem Open Access Chebyshev polynomials on generalized Julia sets(Springer, 2016) Alpan, G.Let (fn)n=1∞ be a sequence of non-linear polynomials satisfying some mild conditions. Furthermore, let Fm(z) : = (fm∘ fm - 1⋯ ∘ f1) (z) and ρm be the leading coefficient of Fm. It is shown that on the Julia set J(fn), the Chebyshev polynomial of degree deg Fm is of the form Fm(z) / ρm- τm for all m∈ N where τm∈ C. This generalizes the result obtained for autonomous Julia sets in Kamo and Borodin (Mosc. Univ. Math. Bull. 49:44–45, 1994). © 2015, Springer-Verlag Berlin Heidelberg.Item Open Access Orthogonal Polynomials Associated with Equilibrium Measures on ℝ(Springer Netherlands, 2017) Alpan, GökalpLet K be a non-polar compact subset of ℝ and μK denote the equilibrium measure of K. Furthermore, let Pn(⋅;μK) be the n-th monic orthogonal polynomial for μK. It is shown that ∥Pn(⋅;μK)∥L2(μK), the Hilbert norm of Pn(⋅;μK) in L2(μK), is bounded below by Cap(K)n for each n∈ ℕ. A sufficient condition is given for(∥Pn(⋅;μK)∥L2(μK)/Cap(K)n)n=1∞ to be unbounded. More detailed results are presented for sets which are union of finitely many intervals. © 2016, Springer Science+Business Media Dordrecht.