Abstract
Let K be a non-polar compact subset of \(\mathbb {R}\) and μ K denote the equilibrium measure of K. Furthermore, let P n (⋅;μ K ) be the n-th monic orthogonal polynomial for μ K . It is shown that \(\|P_{n}\left (\cdot ; \mu _{K}\right )\|_{L^{2}(\mu _{K})}\), the Hilbert norm of P n (⋅;μ K ) in L 2(μ K ), is bounded below by Cap(K)n for each \(n\in \mathbb {N}\). A sufficient condition is given for\(\left (\|P_{n}\left (\cdot ;\mu _{K}\right )\|_{L^{2}(\mu _{K})}/\text {Cap}(K)^{n}\right )_{n=1}^{\infty }\) to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alpan, G., Goncharov, A.: Orthogonal polynomials for the weakly equilibrium Cantor sets. Proc. Amer. Math. Soc. 144(9), 3781–3795 (2016)
Alpan, G., Goncharov, A.: Orthogonal polynomials on generalized Julia sets, Preprint (2015), arXiv:1503.07098v3
Alpan, G., Goncharov, A., Şi̇mşek, A.N.: Asymptotic properties of Jacobi matrices for a family of fractal measures, accepted for publication in Exp. Math.
Aptekarev, A.I.: Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. Mat. Sb. 125, 231–258 (1984). English translations in Math. USSR Sb., 53, 233–260 (1986)
Barnsley, M.F., Geronimo, J.S., Harrington, A.N.: Infinite-dimensional Jacobi matrices associated with Julia sets. Proc. Amer. Math. Soc. 88(4), 625–630 (1983)
Barnsley, M.F., Geronimo, J.S., Harrington, A.N.: Almost periodic Jacobi matrices associated with Julia sets for polynomials. Comm. Math. Phys. 99(3), 303–317 (1985)
Christiansen, J.S.: Szegő’s theorem on Parreau-Widom sets. Adv. Math. 229, 1180–1204 (2012)
Christiansen, J.S., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, II. The Szegö class. Constr. Approx. 33, 365–403 (2011)
Christiansen, J.S., Simon, B., Zinchenko, M.: Asymptotics of Chebyshev Polynomials, I. Subsets of \(\mathbb {R}\), Preprint (2015), arXiv:1505.02604v1
Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math. 171, 1931–2010 (2010)
Geronimo, J.S., Van Assche, W.: Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Amer. Math. Soc. 308, 559–581 (1988)
Goncharov, A., Hatinoğlu, B.: Widom factors. Potential Anal. 42, 671–680 (2015)
Peherstorfer, F.: Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48, 187–205 (1993)
Peherstorfer, F.: Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping. J. Approx. Theory 111, 180–195 (2001)
Peherstorfer, F., Yuditskii, P.: Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89, 113–154 (2003)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press (1995)
Rivlin, T.J.: Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, 2nd edn. Wiley, New York (1990)
Saff, E.B., Totik, V.: Logarithmic potentials with external fields. Springer-Verlag, New York (1997)
Schiefermayr, K.: A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set. East J. Approx. 14, 223–233 (2008)
Simon, B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007)
Simon, B.: Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials. Princeton University Press, Princeton (2011)
Sodin, M., Yuditskii, P.: Functions deviating least from zero on closed subsets of the real axis. St. Petersbg. Math. J. 4, 201–249 (1993)
Stahl, H., Totik, V.: General Orthogonal Polynomials, Encyclopedia of Mathematics, vol. 43. Cambridge University Press, New York (1992)
Totik, V.: Asymptotics for Christoffel functions for general measures on the real line. J. Anal. Math. 81, 283–303 (2000)
Totik, V.: Polynomials inverse images and polynomial inequalities. Acta Math. 187, 139–160 (2001)
Totik, V.: Chebyshev constants and the inheritance problem. J. Approx. Theory 160, 187–201 (2009)
Totik, V.: The polynomial inverse image method. In: Neamtu, M., Schumaker, L. (eds.) Springer Proceedings in Mathematics, Approximation Theory XIII, vol. 13, pp. 345–367. San Antonio (2010)
Totik, V.: Chebyshev polynomials on compact sets. Potential Anal. 40, 511–524 (2014)
Totik, V., Yuditskii, P.: On a conjecture of Widom. J. Approx. Theory 190, 50–61 (2015)
Van Assche, W.: Invariant zero behaviour for orthgonal polynomials on compact sets of the real line. Bull. Soc. Math. Belg. Ser. B 38, 1–13 (1986)
Widom, H.: Polynomials associated with measures in the complex plane. J. Math. Mech. 16, 997–1013 (1967)
Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)
Yudistkii, P.: On the direct cauchy theorem in widom domains: Positive and negative examples. Comput. Methods Funct. Theory 11, 395–414 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by a grant from Tübitak: 115F199.
Rights and permissions
About this article
Cite this article
Alpan, G. Orthogonal Polynomials Associated with Equilibrium Measures on \(\mathbb {R}\) . Potential Anal 46, 393–401 (2017). https://doi.org/10.1007/s11118-016-9589-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-016-9589-3