A Bernstein-Markov theorem for normed spaces

[摘要] Let X and Y be real normed linear spaces and let phi: X --> R be a non-negative function satisfying phi(x + y) less than or equal to phi(x) + //y// for all x, y is an element of X. We show that there exist optimal constants c(m,k) such that if P: X --> Y is any polynomial satisfying //P(x)// less than or equal to phi(x)(m) for all x is an element of X, then //<(D)over cap P-k(x)// less than or equal to c(m,k)phi(x)(m-k) whenever x is an element of X and 0 less than or equal to k less than or equal to m. We obtain estimates for these constants and present applications to polynomials and multilinear mappings in normed spaces. (C) 1997 Academic Press.

[发布日期] 1997-04-15