O'Farrell, A.G. and de Paepe, P.J.
(1993)
Approximation on a disk II.
Mathematische Zeitschrift (212).
pp. 153-156.
ISSN 1432-1823
Abstract
This paper is a continuation of [P]. The main result of [P] is that there are
functions G defined in a neighborhood of the origin in the complex plane, which
behave in a sense as z2, such that G together with z
2 separates the points of (small)
disks D around the origin, and such that the function algebra [z2, G; D] on Dis not
the same as the algebra C(D) of all continuous functions on D. In this paper we
show that the other possibility also can occur: for a large class of functions
G defined in a neighborhood of the origin we show [z2
, G; D] = C(D) for sufficiently
small disks D around 0. We will adopt notation from [P]. In the following it
will be convenient to write the function G in the form
G(z) = z2 (1 + g(z))2
We like to mention that Pascal Thomas, independently from us and at the
same time, worked out a special case of our main result, i.e. the case g(z) = z,
[T].
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