A density result concerning inverse polynomial images
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Abstract:
In this paper, we consider polynomials of degree $n$ for which the inverse image of $[-1,1]$ consists of two Jordan arcs. We prove that the four endpoints of these arcs form an ${\mathcal O}(1/n)$-net in the complex plane.References
- N.I. Achieser, Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen, Bull. Acad. Sci. URSS 7 (1932), 1163–1202.
- A. B. Bogatyrëv, On the efficient computation of Chebyshev polynomials for several intervals, Mat. Sb. 190 (1999), no. 11, 15–50 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 11-12, 1571–1605. MR 1735137, DOI 10.1070/SM1999v190n11ABEH000438
- Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
- Bernd Fischer and Franz Peherstorfer, Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods, Electron. Trans. Numer. Anal. 12 (2001), 205–215. MR 1847918
- S. O. Kamo and P. A. Borodin, Chebyshev polynomials for Julia sets, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1994), 65–67 (Russian); English transl., Moscow Univ. Math. Bull. 49 (1994), no. 5, 44–45 (1995). MR 1318904
- Derek F. Lawden, Elliptic functions and applications, Applied Mathematical Sciences, vol. 80, Springer-Verlag, New York, 1989. MR 1007595, DOI 10.1007/978-1-4757-3980-0
- I. V. Ostrovskii, F. B. Pakovitch, and M. G. Zaidenberg, A remark on complex polynomials of least deviation, Internat. Math. Res. Notices 14 (1996), 699–703. MR 1411590, DOI 10.1155/S1073792896000438
- F. Peherstorfer, Minimal polynomials for compact sets of the complex plane, Constr. Approx. 12 (1996), no. 4, 481–488. MR 1412195, DOI 10.1007/s003659900026
- Franz Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), no. 2, 180–195. MR 1849545, DOI 10.1006/jath.2001.3571
- Franz Peherstorfer and Klaus Schiefermayr, Description of inverse polynomial images which consist of two Jordan arcs with the help of Jacobi’s elliptic functions, Comput. Methods Funct. Theory 4 (2004), no. 2, 355–390. MR 2147391, DOI 10.1007/BF03321075
- Klaus Schiefermayr, Inequalities for the Jacobian elliptic functions with complex modulus, J. Math. Inequal. 6 (2012), no. 1, 91–94. MR 2934569, DOI 10.7153/jmi-06-09
- Klaus Schiefermayr, Inverse polynomial images consisting of an interval and an arc, Comput. Methods Funct. Theory 9 (2009), no. 2, 407–420. MR 2572647, DOI 10.1007/BF03321736
- Klaus Schiefermayr, Geometric properties of inverse polynomial images, Proceedings in Approximation Theory XIII: San Antonio, 2010, Springer Proceedings in Mathematics 13, 2012, pp. 277–287.
- Vilmos Totik, Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), no. 1, 139–160. MR 1864632, DOI 10.1007/BF02392833
Additional Information
- Klaus Schiefermayr
- Affiliation: School of Engineering and Environmental Sciences, University of Applied Sciences Upper Austria, Stelzhamerstr. 23, 4600 Wels, Austria
- Email: klaus.schiefermayr@fh-wels.at
- Received by editor(s): October 24, 2011
- Received by editor(s) in revised form: February 24, 2012, and March 19, 2012
- Published electronically: October 22, 2013
- Communicated by: Walter Van Assche
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 539-545
- MSC (2010): Primary 30C10, 30E10, 33E05, 41A50
- DOI: https://doi.org/10.1090/S0002-9939-2013-11770-3
- MathSciNet review: 3133995