Near-best L1 approximations on circular and elliptical contours

Abstract

Polynomial approximations are obtained to analytic functions on circular and elliptical contours by forming partial sums of order n of their expansions in Taylor series and Chebyshev series of the second kind, respectively. It is proved that the resulting approximations converge in the L 1 norm as n → ∞, and that they are near-best L 1 approximations within relative distances of the order of log n. Practical implications of the results are discussed, and they are shown to provide a theoretical basis for polynomial approximation methods for the evaluation of indefinite integrals on contours.