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Spanning sets of powers on wild Jordan curves

✍ Scribed by Jacob Korevaar; Pia Pfluger

Publisher: Elsevier Science
Year: 1974
Weight: 526 KB
Volume: 77
Category: Article
ISSN: 1385-7258
DOI: 10.1016/1385-7258(74)90020-1

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✦ Synopsis

SIIMMARY

Let y be a Jordan curve in the z-plane which contains the origin in its interior . Every continuous function on y is uniformly approximable by polynomials in z and 1/z (Walsh) . If y is rectifiable, all powers zn are required to obtain a spanning set for C(y), but it has been observed (Warmer) that for nonrectifiable y, the power zo=1 is superfluous . The authors obtain a formula for the distance in C(y) between a given power of a and the closed span of all but a finite number of the other powers (Theorem 2) . This formula leads to various geometric conditions on y under which one can omit a given (finite) number of powers z++, 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) .

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