On the continuity of best polynomial approximations

Abstract:

Suppose f is a continuous complex valued function defined on a compact set E in the plane and ${p_n}(f,E)$ is the polynomial of degree n of best uniform approximation to f on E. If a polynomial ${q_n}$ of degree n approximates f on E “almost” as well as ${p_n}(f,E)$, then ${q_n}$ is “almost” ${p_n}(f,E)$. Sharp estimates, one for the real and one for the general case, are found for ${\left \| {{q_n} - {p_n}(f,E)} \right \|_E}$ in terms of the quantity $({\left \| {f - {q_n}} \right \|_E} - {\left \| {f - {p_n}(f,E)} \right \|_E})$, where ${\left \| \cdot \right \|_E}$ denotes the uniform norm on E.