Abstract
In this paper, we prove a sharp Bernstein-type inequality for algebraic polynomials on circular arcs.
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Bernstein, S.N.: On the Best Approximation of Continuos Functions by Polynomials of Given Degree (O nailuchshem problizhenii nepreryvnykh funktsii posredstrvom mnogochlenov dannoi stepeni. Sobraniye sochinenii, vol. I, pp. 11–104 (1912). Izd. Akad. Nauk SSSR, vol. I (1952), vol. II (1954)
Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer, New York (1995)
Garnett, J.B., Marshall, D.E.: Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2008)
Markov, A.A.: Sur une question posée par D.I. Mendeleieff. Izv. Acad. Nauk St. Petersb. 62, 1–24 (1889)
Nagy, B., Totik, V.: Sharpening of Hilbert’s lemniscate theorem. J. Anal. Math. 96, 191–223 (2005)
Olesov, A.V.: On the application of conformal mappings to inequalities for trigonometric polynomials. Mat. Zametki 76, 396–408 (2004) (Russian). Math. Notes 76, 368–378 (2004)
Peherstorfer, F., Steinbauer, R.: Strong asymptotics of orthonormal polynomials with the aid of Green’s function. SIAM J. Math. Anal. 32, 385–402 (2000)
Pommerenke, Ch.: Boundary Behavior of Conformal Mappings. Grundlehren der mathematischen Wissenschaften, vol. 299. Springer, Berlin (1992)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenschaften, vol. 316. Springer, Berlin (1997)
Riesz, M.: Eine trigonometrische interpolationsformel und einige ungleichungen für polynome. Jahresber. Dtsch. Math.-Ver. 23, 354–368 (1914)
Videnskii, V.S.: Extremal estimates for the derivative of a trigonometric polynomial on an interval shorter than its period. Dokl. Akad. Nauk SSSR 130, 13–16 (1960) (Russian); Sov. Math., Dokl. 1, 5–8 (1960)
Videnskii, V.S.: On trigonometric polynomials of half-integer order. Izv. Akad. Nauk Armjan. SSR Ser. Fiz.-Mat. Nauk 17, 133–140 (1964) (Russian)
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Supported by ERC grant No. 267055.
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Communicated by Doron S. Lubinsky.
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Nagy, B., Totik, V. Bernstein’s Inequality for Algebraic Polynomials on Circular Arcs. Constr Approx 37, 223–232 (2013). https://doi.org/10.1007/s00365-012-9168-9
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DOI: https://doi.org/10.1007/s00365-012-9168-9