Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Inequalities of Chernoff type for finite and infinite sequences of classical orthogonal polynomials

Author(s): Ryszard Smarzewski; Przemysław Rutka
Journal: Proc. Amer. Math. Soc. 138 (2010), 1305-1315.
MSC (2000): Primary 26D10; Secondary 33C45, 60E15
Posted: November 25, 2009
MathSciNet review: 2578524
Retrieve article in: PDF

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Abstract: In this paper we present two-sided estimates of Chernoff type for the weighted $L_{w}^{2}$ -distance of a smooth function to the -dimensional space of all polynomials of degree less than , whenever the weight function solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.

References:

1.: R. P. Agarwal, G. V. Milovanović, Extremal problems, inequalities, and classical orthogonal polynomials, Applied Mathematics and Computation 128 (2002), 151-166. MR 1891017 (2003d:33012)
2.: E. X. L. de Andrade, D. K. Dimitrov, A. Sri Ranga, Characterization of generalized Bessel polynomials in terms of polynomial inequalities, Journal of Mathematical Analysis and Applications 221 (1998), 538-543. MR 1621746 (99b:33011)
3.: G. E. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, 1999. MR 1688958 (2000g:33001)
4.: S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Mathematische Zeitschrift 29 (1929), 730-736. MR 1545034
5.: H. Chernoff, A note on an inequality involving the normal distribution, Annals of Probability 9 (1981), 533-535. MR 614640 (82f:60050)
6.: T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications, 13, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
7.: F. Deutsch, Best Approximation in Inner Product Spaces, Canadian Mathematical Society, Springer-Verlag, New York, 2001. MR 1823556 (2002c:41001)
8.: I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York-London, 1965. MR 0197789 (33:5952)
9.: W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Mathematische Zeitschrift 39 (1935), 634-638. MR 1545524
10.: M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, 98, Cambridge University Press, 2005. MR 2191786 (2007f:33001)
11.: R. Koekoek, R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17 of the Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998.
12.: W. Koepf, M. Masjed-Jamei, A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it, Integral Transforms and Special Functions 17 (2006), 559-576. MR 2246501 (2008b:33021)
13.: H. L. Krall, On derivatives of orthogonal polynomials, Bulletin of the American Mathematical Society 42 (1936), 423-428. MR 1563314
14.: H. L. Krall, On higher derivatives of orthogonal polynomials, Bulletin of the American Mathematical Society 42 (1936), 867-870. MR 1563455
15.: H. L. Krall, On derivatives of orthogonal polynomials. II, Bulletin of the American Mathematical Society 47 (1941), 261-264. MR 0003854 (2:282e)
16.: K. H. Kwon, L. L. Littlejohn, B. H. Yoo, New characterizations of classical orthogonal polynomials, Indagationes Mathematicae 7 (1996), 199-213. MR 1621324 (99g:33026)
17.: P. A. Lesky, Endliche und unendliche Systeme von kontinuierlichen klassischen Orthogonalpolynomen, Zeitschrift für Angewandte Mathematik und Mechanik 76 (1996), 181-184. MR 1382858 (97e:33009)
18.: M. Masjed-Jamei, Classical orthogonal polynomials with weight function $((ax+b)^{2}+ (cx+d)^{2})^{-p}\exp (q {Arctg} ((ax+b)/(cx+d)))$ , $x\in(-\infty,\infty)$ and a generalization of and distributions, Integral Transforms and Special Functions 15 (2004), 137-153. MR 2053407 (2005b:33011)
19.: A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser Verlag, Basel, 1988. MR 922041 (89h:33001)
20.: V. Romanovski, Sur quelques classes nouvelles de polynômes orthogonaux, Comptes Rendus hebdomadaires des séances de l'Académie des sciences 188 (1929), 1023-1025.
21.: V. D. Stepanov, An extremal property of Chebyshev polynomials, Proceedings of the Steklov Institute of Mathematics 248 (2005), 230-242. MR 2165931 (2006h:41006)
22.: V. D. Stepanov, An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives, Proceedings of the American Mathematical Society 136 (2008), 1589-1597. MR 2373588 (2009i:26024)
23.: P. K. Suetin, Classical Orthogonal Polynomials (in Russian), Nauka, Moscow, 1979. MR 548727 (80h:33001)
24.: G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Volume XXIII, Amer. Math. Soc., New York, 1939. MR 0000077 (1:14b)
25.: M. S. Webster, Orthogonal polynomials with orthogonal derivatives, Bulletin of the American Mathematical Society 44 (1938), 880-888. MR 1563895

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