Asymptotics of Chebyshev polynomials, V. residual polynomials

Abstract

We study residual polynomials, R^((e))_(x₀,n), e⊂R, x₀∈R∖e, which are the degree at most n polynomials with R(x₀) = 1 that minimize the supsup norm on ee. New are upper bounds on their norms (that are optimal in some cases) and Szegő–Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case such as root asymptotics, a universal lower bound, and a new characterization of sets saturating this lower bound.

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