In 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree \(n\) with leading coefficient one with respect to the maximum- and the \(L_{1}\)-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.
The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found \(n\)-th root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside \([ - 1, 1]\). But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation \([ - 1, 1]\) remained open. In this paper the author settles this problem with respect to the maximum norm for weight functions whose second derivative is Lip\(\alpha\), \(\alpha \in (0, 1)\), and with respect to the \(L_{1}\)-norm under somewhat stronger differentiability conditions.