Smooth Aggregation Functions on Finite Scales

In this paper smooth aggregation functions on a finite scale are studied and characterized as solutions of a functional equation analogous to the Frank functional equation. The particular cases of quasi-copulas and copulas are also characterized through a similar functional equation. Previous characterizations of these kind of operations through special matrices are used jointly with the new ones to derive some invariant properties on quasi-copulas and copulas on finite scales.

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