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Mappings of Functions

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The following table provides some examples of mappings of real-valued functions. For each mapping it is indicated which of the two conditions of linearity is/are satisfied.

Funktion additivity homogeneity
$ f\mapsto f^\prime$ X X
$ f\mapsto \vert f\vert$ - -
$ f\mapsto \int_0^1 f$ X X
$ f\mapsto \max f$ - -
$ f\mapsto f(0)$ X X
$ f\mapsto (\max f+\min f)/2 $ - X

Observe mapping $ \alpha$ which assigns to each complex-valued function the corresponding real part. This mapping is additive but not homogeneous. Here we have $ \alpha(\mathrm{i}f)\neq \mathrm{i}\alpha(f)$.

(Authors: Burkhardt/Höllig/Hörner)
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  automatisch erstellt am 20.  1. 2005