Extremal Problems for Polynomials with Exponential Weights

For the extremal problem: E„r(a):= min||exp(-W«)(x-+ ■■■)\\L„ a > 0, where U (0 < r < oo) denotes the usual integral norm over R, and the minimum is taken over all monic polynomials of degree n, we describe the asymptotic form of the error E„ r(a) (as n -» oo) 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„(a; x) y„(a)x" + • • • } which are orthonormal on R with respect to exp(-2|A|°). In particular, it is shown that a conjecture of Freud concerning the leading coefficients y „(a) is true in a Cesaro 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 = oo, a > 0 we also prove that, if deg P„(x) < n, the norm ||exp(-|x|")/:'„(x)||i«> is attained on the finite interval [-(«/A0)1/a,(z,A„)1/a], whereAa = r(«)/2-2{r(a/2)}2. Extensions of Nikolskii-type inequalities are also given.