Abstract
Suppose that f is bounded and continuous in a domain D in Rk. Then there exists a best harmonic approximant h to f in the uniform norm. If D is a Jordan domain, f is continuous in D̄, and h is continuous in D̄, then h is unique and can be characterised in terms of the sets in D̄ where h — f assumes the extreme values +m. Examples are given to show that if these hypotheses are relaxed in various ways the conclusion may fail. For instance h need not be continuous in D̄, even if f is continuous in D̄, and if f is only bounded and continuous in D, h need not be unique.
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References
Brelot, M., Sur l’approximation et la convergence dans la théorie des fonctions harmoniques ou holomorphes. Bull. Soc. Math. France 73 (1945), 55–70.
Deny, J., Systèmes totaux de fonctions harmoniques. Ann. Inst. Fourier (Grenoble) 1 (1949), 103–113.
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© 1984 Springer Basel AG
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Hayman, W.K., Kershaw, D., Lyons, T.J. (1984). The Best Harmonic Approximant to a Continuous Function. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_29
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DOI: https://doi.org/10.1007/978-3-0348-5432-0_29
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5434-4
Online ISBN: 978-3-0348-5432-0
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