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Uniform Harmonic Approximation with Continuous Extension to the Boundary

Published online by Cambridge University Press:  20 November 2018

M. Goldstein  and
W. H. Ow
M. Goldstein
Affiliation:
Arizona State University, Tempe, Arizona
W. H. Ow
Affiliation:
Michigan State University, East Lansing, Michigan
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Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such that

uniformly on F as n → ∞.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1375 - 1388
Copyright
Copyright © Canadian Mathematical Society 1988

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