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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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