Bernstein and Nikolskii Inequalities for Erdös Weights

✍ Scribed by T.Z. Mthembu

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 481 KB
• Volume: 75
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1100

✦ Synopsis

Let (W:=e^{-Q}) where (Q) is even, sufficiently smooth, and of faster than polynomial growth at infinity. Such a function (W) is often called an Erdös weight. In this paper we prove Nikolskii inequalities for Erdös weights. We also motivate the usefulness of, and prove a Bernstein inequality of, the form

[
\left.\max {x \in \mathbb{R}}\left|P^{\prime}(x) W(x)\right| 1-\left.\left(\frac{x}{a{\beta} n}\right)^{2}\right|^{x}\left|\leqslant C \frac{n}{a_{n}} \max {x \in \mathbb{R}}\right| P(x) W(x)\left|1-\left(\frac{x}{a{\beta} n}\right)^{2}\right|^{x-1 / 2} \right\rvert,
]