Abstract
Approximation theorems of Walsh imply that the nonnegative integral powers zn, or the integral powers, form a spanning set for certain spaces of functions on compacta in (C. One may ask how many powers may be omitted in various problems of uniform approximation. The present paper surveys “lacunary” approximation theorems for the following kinds of sets: 1. Closed Jordan regions, 2. Jordan arcs, 3. Jordan curves around the origin. At present the most active area is Müntz-type approximation on arcs, in part because of its relation to the Macintyre conjecture for entire functions with gap power series.
Work begun with support from NSF grant MPS 73–08733 at the University of California, San Diego. Second author supported by a grant from the Netherlands research organization ZWO.
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Korevaar, J., Dixon, M. (1978). Lacunary Polynomial Approximation. In: Butzer, P.L., Szökefalvi-Nagy, B. (eds) Linear Spaces and Approximation / Lineare Räume und Approximation. International Series of Numerical Mathematics / Intermationale Schriftenreihe zur Numberischen Mathematik / Sùrie Internationale D’analyse Numùruque, vol 40. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7180-8_42
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