Spanning sets of powers on wild Jordan curves

Abstract

Let γ be a Jordan curve in the z-plane which contains the origin in its interior. Every continuous function on γ is uniformly approximable by polynomials in z and 1 z (Walsh). If γ is rectifiable, all powers z n are required to obtain a spanning set for C(γ), but it has been observed (Wermer) that for nonrectifiable γ, the power z 0 = 1 is superfluous. The authors obtain a formula for the distance in C(γ) between a given power of z and the closed span of all but a finite number of the other powers (Theorem 2). This formula leads to various geometric conditions on γ under which one can omit a given (finite) number of powers z n , and still have a spanning set left (Section 5). The basic tool in the paper is a Walsh type theorem with side conditions (Section 2).