Γ‐Inequalities and Stability of Generalized Extremal Convolutions

Abstract

Convergence properties of convolutions, conjugations, compositions and polarities are expressed in terms of inequalities involving r-operators and generalized extremal convolutions. It is shown that r-inequalities amount to simple algebraic inequalities and (in some cases) to some min-max problems. Numerous applications are indicated, in particular, to continuity properties of the classical conjugation. IntroductionLooking over various reasonings of analysis, one is impressed by a number of arguments that amount to stability of certain connections between mathematical objects. The connections (the stability of which we are going to investigate) are expressed in terms of generalized extremal convolutions (GE-convolutions). They specialize to infimal and supremal convolutions and quasi-convolutions, to conjugations and quasi-conjugations, to compositions of relations and to polarities (Section 1). Consider non empty sets X I , . . . , X , and denote X : = fi X i and X(" := fi X , . i = l i = l , i + kConsider, as well, a function rp : R x R + R, where l? is the extended real line. We define the corresponding generalized infimal convolution (GI-convolution) as the following mapping I: R X x R X + R X :The resulting function is, in fact, an element of but we extend it, by constancy, to the whole of X (In what follows, we shall always tacitly extend considered functions to a common underlying space X). (GS)Symmetrically, the generalized supremal convolution (GS-convolution) is given by scf, 8) := SUP rpcf, g) * x k