L ^sub p^ Christoffel functions, L ^sub p^ universality, and Paley-Wiener spaces

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 125; no. 1; p. 243
• Main Authors: Levin, Eli, Lubinsky, Doron S
• Format: Journal Article
• Language: English
• Published: Jerusalem Springer Nature B.V 01.01.2015
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-015-0008-2

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let [omega] be a regular measure on the unit circle in , and let p > 0. We establish asymptotic behavior, as n[arrow right]∞, for the L p Christoffel function ... at Lebesgue points z on the unit circle in , where [omega]' is lower semi-continuous. While bounds for these are classical, asymptotics have never been established for p [not =] 2. The limit involves an extremal problem in Paley-Wiener space. As a consequence, we deduce universality type limits for the extremal polynomials, which reduce to random-matrix limits involving the sinc kernel in the case p = 2. We also present analogous results for L p Christoffel functions on [-1, 1].