Operations between functions

A structural theory of operations between real-valued (or extended-real-valued) functions on a nonempty subset $A$ of $\mathbb{R}^n$ is initiated. It is shown, for example, that any operation $\ast$ on a cone of functions containing the constant functions, which is pointwise, positively homogeneous, monotonic, and associative, must be one of 40 explicitly given types. In particular, this is the case for operations between pairs of arbitrary, or continuous, or differentiable functions. The term pointwise means that $(f \ast g)(x) = F(f(x), g(x))$, for all $x \in A$ and some function $F$ of two variables. Several results in the same spirit are obtained for operations between convex functions or between support functions. For example, it is shown that ordinary addition is the unique pointwise operation between convex functions satisfying the identity property, i.e., $f \ast 0 = 0 \ast f = f$, for all convex $f$, while other results classify $L_p$ addition. The operations introduced by Volle via monotone norms, of use in convex analysis, are shown to be, with trivial exceptions, precisely the pointwise and positively homogeneous operations between nonnegative convex functions. Several new families of operations are discovered. Some results are also obtained for operations that are not necessarily pointwise. Orlicz addition of functions is introduced and a characterization of the Asplund sum is given. With one exception, a full set of examples is provided showing that none of the assumptions made can be omitted.

Keywords

binary operation, $L_p$ addition, convex function, support function, associativity equation

2010 Mathematics Subject Classification

Primary 26B25, 52A20, 52A41. Secondary 39B22, 39B62.

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First author supported in part by U.S. National Science Foundation Grants DMS-1103612 and DMS-1402929.

Second author supported by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation.

Received 21 November 2015

Published 6 September 2018