Inverse Approximation Theorems of Lebedev and Tamrazov

Abstract

In the period 1966–73, Lebedev and Tamrazov obtained very general inverse approximation theorems for polynomial approximation on plane compacta K. Their work extends the inverse theorems of Dzjadyk for the interval [-1,1] and other well-behaved continua; because of its generality it is rather complicated. The present paper, based on the joint “Master’s thesis” of the first two authors, deals with a simpler situation. It is assumed that K is a continuum with connected complement and that for an f in A(K), the approximation by polynomials p of degree ⩽ n on L = ∂K is at most of order d(x,L1/n)s. Here Lu is the level curve |Φ|= eu of the exterior mapping function Φ and s = k + α, k a nonnegative integer, 0 < α ⩽ < 1. The conclusion is that f is of class ⋀s on L and also on K, that is, f is in Ck and f(k) is in Lip α (if α < 1) or the Zygmund class (if α = 1). Except for integral s this theorem is a very special case of the results of Lebedev and Tamrazov [5], [7].