CHEBYSHEV POLYNOMIALS ON REGULAR POLYGONS IN THE COMPLEX PLANE

Abstract

The Chebyshev problem is studied  for a regular \(m\)-gon \(G_m\), \(m\geq 3\), in the complex plane \(\mathbb{C}\). Let \(\mathscr{P}_n^1\) be the set of algebraic polynomials of a given degree \(n\) with unit leading coefficient. The problem is to find the smallest value \(\tau_n(G_m)\) of the uniform norm \(\|p_n\|_{C(G_m)}\) on the polygon \(G_m\) for polynomials \(p_n\in\mathscr{P}_n^1\), and a polynomial with the smallest norm, called the Chebyshev polynomial (for the polygon). We restrict the class of polynomials in the problem and reduce the domain on which an equivalent problem is studied for polynomials of a substantially lower degree. An exact solution of the problem is given for polynomials of degrees \(n=1,2,\ldots,m\).