Asymptotic Representation of Zolotarev Polynomials

In 1868 Zolotarev determined the polynomial which deviates leastfrom zero with respect to the maximum norm on [–1,1] amongall polynomials of the form where R is given. The polynomial was given explicitly in termsof elliptic functions by Zolotarev. It is now called the Zolotarevpolynomial. Zolotarev also gave an explicit expression for theminimum deviation. In the sequel attempts have been made toreplace the elliptic functions and to express the Zolotarevpolynomial and the minimum deviation in terms of elementaryfunctions, at least asymptotically. In 1913 Bernstein succeededin finding an asymptotic formula for the minimum deviation,which has been improved several times since then. Here we givethe first asymptotic representation of the Zolotarev polynomials.For the asymptotic representation we use the rational functionsintroduced by Bernstein. Furthermore, we obtain asymptotic representationsof minimal polynomials with interpolation constraints whichare of interest in the theory of the iterative solution of inconsistentlinear systems of equations.