Extremal Problems for Polynomials with Exponential Weights

For the extremal problem: \[ {E_{n,r}}(\alpha ): = \min \parallel \exp ( - |x{|^\alpha }) ({x^n} + \cdots ){\parallel _{{L^r}}}, \qquad \alpha > 0,\] 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 $\exp ( - 2|x{|^\alpha })$. 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 $\parallel \exp ( - |x|^{\alpha }) {P_n}(x)\parallel _{{L^\infty }}$ is attained on the finite interval \[ \left [ { - {{(n/{\lambda _\alpha })}^{1/\alpha }},{{(n/{\lambda _\alpha })}^{1/\alpha }}} \right ],\quad {\text {where}}\;{\lambda _\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.\] Extensions of Nikolskii-type inequalities are also given.