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An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives
Author(s):
Vladimir
D.
Stepanov
Journal:
Proc. Amer. Math. Soc.
136
(2008),
1589-1597.
MSC (2000):
Primary 26D10;
Secondary 33C45, 60E15
Posted:
January 4, 2008
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Abstract:
For a weight function generating the classical Jacobi polynomials, the sharp double estimate of the distance from the subspace of all polynomials of an arbitrary fixed order is established.
References:
-
- 1.
- H. Chernoff, A note on an inequality involving the normal distribution, Ann. Probab. 9 (1981), 533-535. MR 614640 (82f:60050)
- 2.
- H. Brascamp and E. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorem, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (1976), 366-389. MR 0450480 (56:8774)
- 3.
- W. Bischoff and M. Fichter, Optimal lower and upper bounds for the
 -mean deviation of functions of a random variable, Math. Methods in Statisties. 9 (2000), 237-269. MR 1807094 (2001k:60022) - 4.
- V. D. Stepanov, An extremal property of Chebyshev polynomials, Proc. Steklov Inst. Math. 248 (2005), 230-242. MR 2165931 (2006h:41006)
- 5.
- P. K. Suetin, Classical orthogonal polynomials (in Russian), Nauka, Moscow, 1979. MR 548727 (80h:33001)
- 6.
- G. Aleksich, The convergence problems of orthogonal series (in Russian), Izd. Inostr. Lit., Moscow, 1963. MR 0218828 (36:1912)
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Additional Information:
Vladimir
D.
Stepanov
Affiliation:
People Friendship University, Miklukho-Maklai 6, Moscow, 117198, Russia
Email:
vstepanov@sci.pfu.edu.ru
DOI:
10.1090/S0002-9939-08-09218-6
PII:
S 0002-9939(08)09218-6
Keywords:
Jacobi polynomials,
Chernoff inequality
Received by editor(s):
March 4, 2006
Posted:
January 4, 2008
Additional Notes:
The work of the author was financially supported by the Russian Foundation for Basic Researches (Projects 05--01--00422, 06--01--00341, 06--01--04006 and 07--01--00054) and by the INTAS grant 05-1000008-8157.
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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