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Markov–Bernstein and Nikolskiui Inequalities, and Christoffel Functions for Exponential Weights on $( - 1,1)$

1993; Society for Industrial and Applied Mathematics; Volume: 24; Issue: 2 Linguagem: Inglês

10.1137/0524033

1095-7154

D. S. Lubinsky, E. B. Saff,

Analytic and geometric function theory

Exponential weights $w: = e^{ - Q} $ are considered, where $Q:( - 1,1) \to \bf {R}$ convex, and sufficiently smooth. For example, the results may be applied to \[\begin{gathered} w(x): = (1 - x^2 )^\alpha ,\quad \alpha > 0, \hfill \\ w(x): = \exp ( - (1 - x^2 )^{ - \alpha } ),\quad \alpha > 0,\quad {\text{or}} \hfill \\ w(x): = \exp ( - \exp _k (1 - x^2 )^{ - \alpha } ),\quad \alpha > 0,\quad \,k \geq 1, \hfill \\ \end{gathered} \] where $\exp _k = \exp (\exp ( \cdots ))$ denotes the kth iterated exponential. Weighted Markov and Bernstein inequalities such as \[\left\|P'w\right\|_{L_\infty [ - 1,1]} \leq CQ'(a_{2n} )\|Pw\|_{L_\infty [ - 1,1]} ,\] and \[ \|P' w\| \leq \frac{{Cn}}{{\sqrt {1 - {{ |x| } / {a_n }}} }} \|Pw\|_{L_\infty [ - 1,1]} ,\quad |x| < a_n ,\] are established for polynomials P of degree at most n. Here an is the $a_n$th Mhaskar–Rahmanov–Saff number for Q. For the special weights listed above, a more explicit form is given to $Q'(a_{2n} )$. Estimates are deduced for ChristofFel functions such as \[\mathop {\sup }\limits_{x \in [ - 1,1]} \lambda _n^{ - 1} (w^2 ,x)w(x) \leq CQ'(a_{2n} ),\] and also Nikolskii inequalities.