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Extremal problems for polynomials with exponential weights


Authors: H. N. Mhaskar and E. B. Saff
Journal: Trans. Amer. Math. Soc. 285 (1984), 203-234
MSC: Primary 41A17; Secondary 42C05
DOI: https://doi.org/10.1090/S0002-9947-1984-0748838-0
MathSciNet review: 748838
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Abstract: For the extremal problem:

$\displaystyle {E_{n,r}}(\alpha ): = \min \parallel \exp ( - \vert x{\vert^\alpha })\,({x^n} + \cdots ){\parallel_{{L^r}}}, \qquad \alpha > 0,$

where $ {L^r}\,(0 < r \leqslant \infty )$ denotes the usual integral norm over $ {\mathbf{R}}$, and the minimum is taken over all monic polynomials of degree $ n$, we describe the asymptotic form of the error $ {E_{n,r}}(\alpha )\;({\text{as}}\;n \to \infty )$ as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case $ r = 2$ yields new information regarding the polynomials $ \{ {p_n}(\alpha ;x) = {\gamma_n}(\alpha )\,{x^n} + \cdots \} $ which are orthonormal on $ {\mathbf{R}}$ with respect to $ \exp ( - 2\vert x{\vert^\alpha })$. In particular, it is shown that a conjecture of Freud concerning the leading coefficients $ {\gamma_n}(\alpha )$ is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud's conjecture. For $ r = \infty ,\alpha > 0$ we also prove that, if $ \deg {P_n}(x) \leqslant n$, the norm $ \parallel \exp ( - \vert x\vert^{\alpha })\,{P_n}(x)\parallel_{{L^\infty }}$ is attained on the finite interval

$\displaystyle \left[ { - {{(n/{\lambda_\alpha })}^{1/\alpha }},{{(n/{\lambda_\a... ...ambda_\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.$

Extensions of Nikolskii-type inequalities are also given.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0748838-0
Keywords: Orthogonal polynomials, exponential weights, error estimates, zero distribution, Nikolskii inequalities
Article copyright: © Copyright 1984 American Mathematical Society

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