A complex extremal problem of Chebyshev type
Consider the class \({\mathcal P}_n\) of all complex polynomials of fixed degree \(n\) which are bounded by 1 on the interval \([-1,1]\). What can we say about \(\sup\{| p(z_0) |:p\in {\mathcal P}_n\}= :L(z_0)\)? For \(z_0\in \mathbb{R} \setminus]-1,1[\) it is a classical result that \(L(z_0)\) is given by the modulus of the \(n\)th degree Chebyshev polynomial. The case \(\mathfrak R\mathfrak z_0=0\) has been studied by \textit{R. Freund} and \textit{St. Ruscheweyh} [Numer. Math. 48, 525-542 (1986; Zbl 0611.65016)]. The main subject of this paper is to solve this extremal problem for the remaining points \(z_0\). An explicit expression in terms of elliptic functions is given.
scientific article; zbMATH DE number 1309904