On approximation of affine Baire-one functions. (English) Zbl 1031.35011

This paper is a significant contribution to the important area of interaction between potential theory and convex analysis. Its purpose is to explore possible generalizations of known results concerning Baire-one affine functions on a compact convex set \(X\) to the context of general function spaces \({\mathcal H}\). A key motivating and guiding example throughout is the case where \({\mathcal H}\) is the collection of all harmonic functions an a bounded open set \(U\subset \mathbb{R}^m\) that are continuous on \(\overline U\). This “harmonic case” is used to identify some properties which fail to generalize to the function space context and to formulate new general results, especially for simplicial function spaces. Several characterizations are given of those Baire-one functions which are pointwise limits of a bounded sequence of elements from \({\mathcal H}\). It is shown that, in general, the pointwise limits of \({\mathcal H}\)-affine functions need not coincide with the Baire-one \({\mathcal H}\)-affine functions. However, when \({\mathcal H}\) is simplicial, each bounded Baire-one \({\mathcal H}\)-affine function is the pointwise limit of a bounded sequence of continuous \({\mathcal H}\)-affine functions. The paper is enriched by several illuminating and clever constructions of counterexamples, mainly in the harmonic case.


35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
46A55 Convex sets in topological linear spaces; Choquet theory
Full Text: DOI


[1] Alfsen, E. M., Compact Convex Sets and Boundary Integrals (1971), Berlin: Springer-Verlag, Berlin · Zbl 0209.42601
[2] Ancona, A., Démonstration d’une conjecture sur la capacité et l’effilement, Comptes Rendus de l’Académie des Sciences, Paris, 297, 393-395 (1983) · Zbl 0544.31006
[3] Armitage, D. H.; Gardiner, S. J., Classical Potential Theory (2001), Berlin: Springer-Verlag, Berlin · Zbl 0972.31001
[4] Asimow, L.; Ellis, A. J., Convexity Theory and its Applications in Functional Analysis (1980), New York: Academic Press, New York · Zbl 0453.46013
[5] Bauer, H., Minimalstellen von Funktionen und Extremalpunkte, Archiv der Mathematik, 9, 389-393 (1958) · Zbl 0082.32601
[6] Bauer, H., Šilowscher Rand und Dirichletsches Problem, Annales de l’Institut Fourier (Grenoble), 11, 89-136 (1961) · Zbl 0098.06902
[7] H. Bauer,Konvexität in topologischen Vektorräumen, Universität Hamburg, 1963/64.
[8] Bauer, H., Simplicial function spaces and simplexes, Expositiones Mathematicae, 3, 165-168 (1985) · Zbl 0564.46007
[9] Bishop, E.; de Leeuw, K., The representation of linear functionals by measures on sets of extreme points, Annales de l’Institut Fourier (Grenoble), 9, 305-331 (1959) · Zbl 0096.08103
[10] Bliedtner, J.; Hansen, W., Simplicial cones in potential theory, Inventiones Mathematicae, 29, 83-110 (1975) · Zbl 0308.31011
[11] Bliedtner, J.; Hansen, W., Potential Theory—An Analytic and Probabilistic Approach to Balayage (1986), Berlin: Springer-Verlag, Berlin · Zbl 0706.31001
[12] Boboc, N.; Cornea, A., Convex cones of lower semicontinuous functions on compact spaces, Revue Roumaine de Mathématiques Pures et Appliquées, 12, 471-525 (1967) · Zbl 0155.17301
[13] Brelot, M., On Topologies and Boundaries in Potential Theory (1971), Berlin: Springer-Verlag, Berlin · Zbl 0222.31014
[14] G. Choquet,Remarque à propos de la démonstration de l’unicité de P. A Meyer, Séminaire Brelot-Choquet-Deny (Théorie de Potentiel)6 (1961/62), Exposé No. 8. · Zbl 0115.32402
[15] Choquet, G., Lectures on Analysis I-III (1969), New York-Amsterdam: W. A. Benjamin, Inc., New York-Amsterdam
[16] Constantinescu, C.; Cornea, A., Potential Theory on Harmonic Spaces (1972), Berlin: Springer-Verlag, Berlin · Zbl 0248.31011
[17] Dellacherie, C.; Meyer, P. A., Probabilités et Potentiel (1987), Paris: Hermann, Paris · Zbl 0624.60084
[18] Edwards, D. A., Minimum-stable wedges of semicontinuous functions, Mathematica Scandinavica, 19, 15-26 (1966) · Zbl 0146.37002
[19] Effros, E. G.; Kazdan, J. L., Applications of Choquet simplexes to elliptic and parabolic boundary value problems, Journal of Differential Equations, 8, 95-134 (1970) · Zbl 0255.46018
[20] Fuglede, B., Remarks on fine continuity and the base operation in potential theory, Mathematische Annalen, 210, 207-212 (1974) · Zbl 0273.31014
[21] S. J. Gardiner,Harmonic Approximation, Cambridge University Press, 1995. · Zbl 0826.31002
[22] Hartogs, F.; Rosenthal, A., Über Folgen analytischer Funktionen, Mathematische Annalen, 100, 212-263 (1928) · JFM 54.0358.01
[23] Lazar, A., Spaces of affine continuous functions on simplexes, Transactions of the American Mathematical Society, 134, 503-525 (1968) · Zbl 0174.17102
[24] Lukeš, J.; Malý, J.; Zajíček, L., Fine Topology Methods in Real Analysis and Potential Theory (1986), Berlin: Springer-Verlag, Berlin · Zbl 0607.31001
[25] Netuka, I., The Dirichlet problem for harmonic functions, The American Mathematical Monthly, 87, 621-628 (1980) · Zbl 0454.31002
[26] Osgood, W. F., Note on the functions defined by infinite series whose terms are analytic functions, Annals of Mathematics, 3, 25-34 (1901) · JFM 32.0399.01
[27] Phelps, R. R., Lectures on Choquet’s Theorem (1966), New York: Van Nostrand, New York · Zbl 0135.36203
[28] Raymond, J. Saint, Fonctions convexes de première classe, Mathematica Scandinavica, 54, 121-129 (1984) · Zbl 0599.46012
[29] Rogalski, M., Opérateurs de Lion, projecteurs boréliens et simplexes analytiques, Journal of Functional Analysis, 2, 458-488 (1968) · Zbl 0164.43403
[30] Roy, N. M., Extreme points of convex sets in infinite dimensional spaces, The American Mathematical Monthly, 94, 409-422 (1987) · Zbl 0664.46006
[31] Štěpničková, L., Pointwise and locally uniform convergence of holomorphic and harmonic functions, Commentationes Mathematicae Universitatis Carolinae, 40, 665-678 (1999) · Zbl 1009.31002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.