On polynomials that deviate least from zero in the metric of \(L[-1,1]\).
Let \(l\), \(n\) be positive integers and \({A_0=1}\), \({A_1,\dots ,A_{n-1}}\) be real numbers. It is required to find real numbers \({a_l,a_{l+1},\dots,a_{l+n}}\) minimizing the value \[ \int^{+1}_{-1}| x^{n+l}+A_1x^{n+l-1}+\cdots+a_lx^{n}+\dots+a_{n+l}| \,dx. \] This is the classical E.I.~Zolotarev problem in the general statement for the space \({L_1[-1,+1]}\). At \({l=1}\), the problem was solved by Korkin and Zolotarev in 1873. \textit{E.~Galeev} solved the problem for \({l=2}\) [Math. Notes 17, 9--13 (1975; Zbl 0321.41019)]. In the paper under review, Geit suggests a new technique for solving the problem and obtains explicit solutions for the cases \({l=3}\) and \({l=4}\).
scientific article; zbMATH DE number 2074510