Notes on inequalities with doubling weights

JOURNAL OF APPROXIMATION THEORY

卷:100

Article

Erdélyi, T

关键词: Bernstein's inequality;    Marcinkiewicz inequality;    Christoffel function;    Remez inequality;    Nikolskii inequality;    doubling weights;    A alpha weights;   

DOI  :  10.1006/jath.1999.3340

来源: Elsevier

【 摘 要 】

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii. Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most ut. the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly stronger A(infinity) condition is used. Throughout their paper the L-p norm is studied under the assumption 1 less than or equal to p < infinity. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0 < p < 1. The crucial tool it, an estimate for quadrature sums for the pth power (0 < p < infinity is arbitrary) of trigonometric polynomials established by Lubinsky, Miti, and Nevai. For technical reasons we discuss only the trigonometric cases. (C) 1999 Academic Press.

【 授权许可】

Free