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- math-csgb-2015-11
Final published version, 954 KB, PDF document

- http://math.au.dk/publs?publid=1049
Final published version

- Richard J. Gardner, Department of Mathematics, Western Washington University, United States
- Markus Kiderlen

A structural theory of operations between real-valued (or extended-real-valued) functions on a nonempty subset A of R^{n}
is initiated. It is shown, for example,
that any operation ∗ 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 ∗g)(x) = F(f(x), g(x)), for all x ∈ 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 ∗ 0 = 0 ∗ 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. A full set of examples is provided showing that none
of the assumptions made can be omitted.

Original language | English |
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Publisher | Centre for Stochastic Geometry and advanced Bioimaging, Aarhus University |

Number of pages | 53 |

Publication status | Published - 2015 |

Series | CSGB Research Reports |
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Number | 11 |

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ID: 97725183