The best extension of algebraic polynomials from the unit circle

Abstract

We study the class \( \mathfrak{P}_n \) of algebraic polynomials Pn(x, y) in two variables of total degree n whose uniform norm on the unit circle Γ1 centered at the origin is at most 1: \( \left\| {P_n } \right\|_{C(\Gamma _1 )} \) ≤ 1. The extension of polynomials from the class \( \mathfrak{P}_n \) to the plane with the least uniform norm on the concentric circle Γr of radius r is investigated. It is proved that the values θn(r) of the best extension of the class \( \mathfrak{P}_n \) satisfy the equalities θn(r) = rn for r > 1 and θn(r) = rn−1 for 0 < r < 1.