Asymptotic Behavior of Nikolskii Constants for Polynomials on the Unit Circle

Let q > p > 0 , and consider the Nikolskii constants Λ n , p , q = inf deg P ≤ n - 1 P p P q , where the norm is with respect to normalized Lebesgue measure on the unit circle. We prove that lim sup n → ∞ n 1 p - 1 q Λ n , p , q ≤ E p , q , where E p , q = inf f L p R f L q R , and the inf is...

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Bibliographic Details
Published inComputational methods and function theory Vol. 15; no. 3; pp. 459 - 468
Main Authors Levin, Eli, Lubinsky, Doron S.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2015
Subjects
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Nikolskii Inequalities
42C05
Paley–Wiener Spaces
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Summary:Let q > p > 0 , and consider the Nikolskii constants Λ n , p , q = inf deg P ≤ n - 1 P p P q , where the norm is with respect to normalized Lebesgue measure on the unit circle. We prove that lim sup n → ∞ n 1 p - 1 q Λ n , p , q ≤ E p , q , where E p , q = inf f L p R f L q R , and the inf is taken over all entire functions f of exponential type at most π . We conjecture that the lim sup can be replaced by a limit.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-015-0113-3