Extremal problems for polynomials with exponential weights

For the extremal problem: \[ E n , r ( α ) := min ∥ exp ⁡ ( − | x | α ) ( x n + ⋯ ) ∥ L r , α > 0 , {E_{n,r}}(\alpha ): = \min \parallel \exp ( - |x{|^\alpha })\,({x^n} + \cdots ){\parallel _{{L^r}}}, \qquad \alpha > 0, \] where L r ( 0 > r ⩽ ∞ ) {L^r}\,(0 > r \leqslant \infty ) denotes the usual integral norm over R {\mathbf {R}} , and the minimum is taken over all monic polynomials of degree n n , we describe the asymptotic form of the error E n , r ( α ) ( as n → ∞ ) {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 r = 2 yields new information regarding the polynomials { p n ( α ; x ) = γ n ( α ) x n + ⋯ } \{ {p_n}(\alpha ;x) = {\gamma _n}(\alpha )\,{x^n} + \cdots \} which are orthonormal on R {\mathbf {R}} with respect to exp ⁡ ( − 2 | x | α ) \exp ( - 2|x{|^\alpha }) . In particular, it is shown that a conjecture of Freud concerning the leading coefficients γ n ( α ) {\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 = ∞ , α > 0 r = \infty ,\alpha > 0 we also prove that, if deg ⁡ P n ( x ) ⩽ n \deg {P_n}(x) \leqslant n , the norm ∥ exp ⁡ ( − | x | α ) P n ( x ) ∥ L ∞ \parallel \exp ( - |x|^{\alpha })\,{P_n}(x)\parallel _{{L^\infty }} is attained on the finite interval \[ [ − ( n / λ α ) 1 / α , ( n / λ α ) 1 / α ] , where λ α = Γ ( α ) / 2 α − 2 { Γ ( α / 2 ) } 2 . \left [ { - {{(n/{\lambda _\alpha })}^{1/\alpha }},{{(n/{\lambda _\alpha })}^{1/\alpha }}} \right ],\quad {\text {where}}\;{\lambda _\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}. \] Extensions of Nikolskii-type inequalities are also given.