On a Generalized Infimal Convolution of Set Functions

• Autorzy: Gergely Pataki
• Języki publikacji: EN

Abstrakty

EN

Having in mind the ideas of J. Moreau, T. Strömberg and Á. Száz, for any function $f$ and $g$ of one power set $\mathcal{P}(X)$ to another $\mathcal{P}(Y)$, we define an other function $(f*g)$ of $\mathcal{P}(X)$ to $\mathcal{P}(Y)$ such that $$(f*g)(A) = \bigcap \{ f(U) \cup g(V) : A \subset U \cup V \subset X \}$$ for all $A \subset X$. Thus $(f*g)$ is a generalized infimal convolution of $f$ and $g$. We show that if $f$ and $g$ preserve arbitrary unions, then $(f*g)$ also preserves arbitrary unions. Moreover, if $F$ and $G$ are relations on $X$ to $Y$ such that $$F(x) = f(\{x\}) \ and \ G(x) = g(\{x\})$$ for all $x \in X$, then $$(f*g)(A) = (F \cap G)[A]$$ for all $A \subset X$. (original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical functions

PL

Matematyka; Funkcje matematyczne

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2013
• Tom: 27
• Strony: 99--106

Twórcy

autor

Gergely Pataki

• Budapest University of Technology and Economics, Hungary

Bibliografia

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• Glavosits T., Száz Á., The generalized infimal convolution can be used to naturally prove some dominated monotone additive extension theorems, Annales Mathematicae Silesianae 25 (2011), 67-100.
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• Höhle U., Kubiak T., On regularity of sup-preserving maps: generalizing Zareckli ̆ı's theorem, Semigroup Forum 83 (2011), 313-319.
• Moreau J.J., Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. 49 (1970), 109-154.
• Strömberg T., The operation of infimal convolution, Dissertationes Math. 352 (1996), 1-58.
• Száz Á., The intersection convolution of relations and the Hahn-Banach type theorems, Ann. Polon. Math. 69 (1998), 235-249.
• Száz Á., The infimal convolution can be used to derive extension theorems from the sandwich ones, Acta Sci. Math. (Szeged) 76 (2010), 489-499.
• Száz Á., A reduction theorem for a generalized infimal convolution, Tech. Rep., Inst. Math. Inf., Univ. Debrecen 11 (2009), 1-4.