Dense Markov Spaces and Unbounded Bernstein Inequalities

✍ Scribed by P. Borwein; T. Erdelyi

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 323 KB
• Volume: 81
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1033

✦ Synopsis

An infinite Markov system (\left{f_{0}, f_{1}, \ldots\right}) of (C^{2}) functions on ([a, b]) has dense span in (C[a, b]) if and only if there is an unbounded Bernstein inequality on every subinterval of ([a, b]). That is if and only if, for each ([\alpha, \beta]=[a, b], \alpha \neq \beta), and (\gamma>0), we can find (g \in \operatorname{span}\left{f_{0}, f_{1}, \ldots\right}) with (\left|g^{\prime}\right|{[\alpha, \beta]}>y^{\prime}|g|{[a . h]}). This is proved under the assumption (\left(f_{1} / f_{0}\right)^{\prime}) does not vanish on ((a, b)). Extension to higher derivatives are also considered. An interesting consequence of this is that functions in the closure of the span of a non-dense (C^{2}) Markov system are always (C^{n}) on some subinterval. (i) 1995 Academic Press. Inc.