Bernstein classes

On Bernstein Classes of Quasianalytic Maps

We study the structure of families of “well approximable” elements of tensor products of Banach spaces including analogs of the classical quasianalytic classes in the sense of Bernstein and Beurling. As in the case of quasianalytic functions, we prove for members of these families variants of the Mazurkiewicz and Markushevich theorems and in some particular cases, if such elements are Banach-va...

MARKOV AND BERNSTEIN TYPE INEQUALITIES IN Lp FOR CLASSES OF POLYNOMIALS WITH CONSTRAINTS

The Markov-type inequality i \Ax)\dx is proved for all real algebraic polynomials/of degree at most n having at most k, with 0 ^ k < n, zeros (counting multiplicities) in the open unit disk of the complex plane, and for all p > 0, where c(p) = c(l +/T) with some absolute constant c> 0. This inequality has been conjectured since 1983 when the L^ case of the above result was proved. It improves a...

Exact Values of Bernstein n-Widths for Some Classes of Convolution Functions

Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator

Guo Feng School of Mathematics and Information Engineering, Taizhou University, Taizhou 317000, China Correspondence should be addressed to Guo Feng, [email protected] Received 9 May 2011; Accepted 2 August 2011 Academic Editor: Kai Diethelm Copyright q 2012 Guo Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distri...

Markov-Bernstein Type Inequalities for Classes of Polynomials with Restricted Zeros

We prove that there exists an absolute constant c > 0 such that |p′(y)| ≤ c min { n(k + 1), ( n(k + 1) 1 − y2 )1/2} max −1≤x≤1 |p(x)|, −1 ≤ y ≤ 1 for every real algebraic polynomial of degree at most n having at most k zeros in the open unit disk {z ∈ C : |z| < 1}. This inequality, which has been conjectured for at least a decade, improves and generalizes several earlier results. Up to the mult...