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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.

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
46A55 Convex sets in topological linear spaces; Choquet theory
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[1] E. M. Alfsen,Compact Convex Sets and Boundary Integrals, Springer-Verlag, Berlin, 1971. · Zbl 0209.42601
[2] A. Ancona,Démonstration d’une conjecture sur la capacité et l’effilement, Comptes Rendus de l’Académie des Sciences, Paris297 (1983), 393–395.
[3] D. H. Armitage and S. J. Gardiner,Classical Potential Theory, Springer-Verlag, Berlin, 2001.
[4] L. Asimow and A. J. Ellis,Convexity Theory and its Applications in Functional Analysis, Academic Press, New York, 1980. · Zbl 0453.46013
[5] H. Bauer,Minimalstellen von Funktionen und Extremalpunkte, Archiv der Mathematik9 (1958), 389–393. · Zbl 0082.32601
[6] H. Bauer,Šilowscher Rand und Dirichletsches Problem, Annales de l’Institut Fourier (Grenoble)11 (1961), 89–136.
[7] H. Bauer,Konvexität in topologischen Vektorräumen, Universität Hamburg, 1963/64.
[8] H. Bauer,Simplicial function spaces and simplexes, Expositiones Mathematicae3 (1985), 165–168. · Zbl 0564.46007
[9] E. Bishop and K. de Leeuw,The representation of linear functionals by measures on sets of extreme points, Annales de l’Institut Fourier (Grenoble)9 (1959), 305–331. · Zbl 0096.08103
[10] J. Bliedtner and W. Hansen,Simplicial cones in potential theory, Inventiones Mathematicae29 (1975), 83–110. · Zbl 0308.31011
[11] J. Bliedtner and W. Hansen,Potential Theory–An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, Berlin, 1986. · Zbl 0706.31001
[12] N. Boboc and A. Cornea,Convex cones of lower semicontinuous functions on compact spaces, Revue Roumaine de Mathématiques Pures et Appliquées12 (1967), 471–525. · Zbl 0155.17301
[13] M. Brelot,On Topologies and Boundaries in Potential Theory, Lecture Notes in Mathematics175, Springer-Verlag, Berlin, 1971. · 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.
[15] G. Choquet,Lectures on Analysis I–III, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0181.39603
[16] C. Constantinescu and A. Cornea,Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin, 1972. · Zbl 0248.31011
[17] C. Dellacherie and P. A. Meyer,Probabilités et Potentiel, Hermann, Paris, 1987.
[18] D. A. Edwards,Minimum-stable wedges of semicontinuous functions, Mathematica Scandinavica19 (1966), 15–26. · Zbl 0146.37002
[19] E. G. Effros and J. L. Kazdan,Applications of Choquet simplexes to elliptic and parabolic boundary value problems, Journal of Differential Equations8 (1970), 95–134. · Zbl 0255.46018
[20] B. Fuglede,Remarks on fine continuity and the base operation in potential theory, Mathematische Annalen210 (1974), 207–212. · Zbl 0284.31011
[21] S. J. Gardiner,Harmonic Approximation, Cambridge University Press, 1995. · Zbl 0826.31002
[22] F. Hartogs and A. Rosenthal,Über Folgen analytischer Funktionen, Mathematische Annalen100 (1928), 212–263. · JFM 54.0358.01
[23] A. Lazar,Spaces of affine continuous functions on simplexes, Transactions of the American Mathematical Society134 (1968), 503–525. · Zbl 0174.17102
[24] J. Lukeš, J. Malý and L. Zajíček,Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics1189, Springer-Verlag, Berlin, 1986.
[25] I. Netuka,The Dirichlet problem for harmonic functions, The American Mathematical Monthly87 (1980), 621–628. · Zbl 0454.31002
[26] W. F. Osgood,Note on the functions defined by infinite series whose terms are analytic functions, Annals of Mathematics3 (1901), 25–34. · JFM 32.0399.01
[27] R. R. Phelps,Lectures on Choquet’s Theorem, Van Nostrand Mathematics Studies No. 7, Van Nostrand, New York, 1966. · Zbl 0135.36203
[28] J. Saint Raymond,Fonctions convexes de première classe, Mathematica Scandinavica54 (1984), 121–129. · Zbl 0599.46012
[29] M. Rogalski,Opérateurs de Lion, projecteurs boréliens et simplexes analytiques, Journal of Functional Analysis2 (1968), 458–488. · Zbl 0164.43403
[30] N. M. Roy,Extreme points of convex sets in infinite dimensional spaces, The American Mathematical Monthly94 (1987), 409–422. · Zbl 0664.46006
[31] L. Štěpničková,Pointwise and locally uniform convergence of holomorphic and harmonic functions, Commentationes Mathematicae Universitatis Carolinae40 (1999), 665–678. · Zbl 1009.31002
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