\(L_{\infty}\) Markov and Bernstein inequalities for Erdős weights
The results of this remarkable paper can be reviewed best possible way by recalling the summary: ``Recently, weighted Markov and Bernstein inequalities have been established for large classes of Freud weights, that is, weights of the form \(W(x):=e^{-Q(x)}\), where Q(x) is even and of smooth polynomial growth at infinity. In this paper, we consider Erdős weights, which have the form \(W(x):=e^{-Q(x)}\), where Q(x) is even and of faster than polynomial growth at infinity. For a large class of Erdős weights, we establish the Markov type inequality \((1)\quad \| P'W\|_{{\mathbb{R}}}\leq CQ'(a_ n)\| PW\|_{{\mathbb{R}}},\) for \(n\geq 1\) and P any polynomial of degree at most n. Here the norm is the sup norm, and C is independent of n and P, while \(a_ n\) is the Mhaskar- Rahmanov-Saff number, that is, it is the positive root of the equation \((2)\quad n=\frac{2}{\pi}\int^{1}_{0}a_ ntQ'(a_ nt)dt/\sqrt{1- t^ 2}.\) For example, we consider \(Q(x):=\exp_ k(| x|^{\alpha})\), where \(\alpha >0\), and where \(\exp_ k\) denotes the kth iterated exponential, and give a more explicit formulation of (1). We also establish Bernstein type inequalities that for part of the range (- \(\infty,\infty)\) improve on (1).''
scientific article; zbMATH DE number 4146766
scientific article