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Representation of functions of several variables by difference of convex functions. (English. Russian original) Zbl 0906.26006
J. Math. Sci., New York 100, No. 3, 2209-2227 (2000); translation from Zalgaller, V. A. (ed.) et al., Geometry and topology. 2. Work collection. Sankt-Peterburg: Matematicheskij Institut Im. V. A. Steklova, Sankt-Peterburgskoe Otdelenie, RAN, Zap. Nauchn. Semin. POMI. 246, 36-65 (1997).
Summary: If a function \(f: D^n\to\mathbb{R}\), where \(D^n\) is a convex compact set in \(\mathbb{R}^n\), admits a decomposition \(f= g-h\) with convex \(g\), \(h\), where \(h\) is upper bounded, then there exists such a decomposition which is in some sense “minimal”. A recurrent procedure converging to that decomposition is given. For piecewise linear functions \(f\), finite algorithms of those decompositions for \(n= 1,2\) are given. A number of examples clarifying some unexpected effects is presented and problems are formulated.

26B40 Representation and superposition of functions
26B25 Convexity of real functions of several variables, generalizations
Full Text: DOI
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