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Weighted Markov‐type inequalities, norms of Volterra operators, and zeros of Bessel functions
Author(s) -
Böttcher Albrecht,
Dörfler Peter
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810274
Subject(s) - mathematics , bessel function , laguerre polynomials , norm (philosophy) , operator norm , pure mathematics , markov chain , volterra integral equation , weight function , order (exchange) , operator (biology) , dual norm , mathematical analysis , operator theory , integral equation , banach space , statistics , biochemistry , chemistry , finance , repressor , political science , transcription factor , law , economics , gene
The first term of the asymptotics of the best constants in Markov‐type inequalities for higher derivatives of polynomials is determined in the two cases where the underlying norm is the L 2 norm with Laguerre weight or the L 2 norm with Gegenbauer weight. The coefficient in this term is shown to be the norm of a certain Volterra integral operator which depends on the weight and the order of the derivative. For first order derivatives, the norms of the Volterra operators are expressed in terms of the zeros of Bessel functions. The asymptotic behavior of the coefficients is studied and tight bounds for them are given (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)