Some inequalities for algebraic polynomials with the Laguerre weight

[摘要] There is a series of publications which have considered inequalities of Markov-Bemstein-Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159-176; B.D. Bojanov, An extension of the Markov inequality. J. Approx. Theory 35 (1982) 181-190; P. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov-Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. I 0 972) 15-25: A. Guessab, GX Milovanovic, Weighted L-2-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244-249-,I.I. Ibragimov, Some inequalities for algebraic polynomials. in: V.I. Smirnov (Ed.), Fizmatgiz. 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570-572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) 0 957) 32-35; M. K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Uber die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehorige Extremum, Math. Ann. 119 (1944) 165-209; P. TurAn, Remark on a theorem of Erhard Schmidt, Mathernatica 2 (25) (1960) 373-378]). In this paper we find an inequality of the same type for algebraic polynomials on (0, infinity) with the Laguerre weight function e(-x)x(-alpha) (alpha > - 1). (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-01