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Given the functions $f\colon A\to B$ and $g\colon B\to B$, a common, useful strategy is to define a new function $h\colon A\to A$ as the composition $f^{-1}\circ g\circ f$.

There seem to be many applications and conceptually similar maneuvers, ranging from the "Schwarzian transform" (used in programming to sort arbitrary data without redundant key computation) to Fourier convolution ($f * g = \mathcal{F}^{-1}\{\mathcal{F}\{f\}\cdot \mathcal{F}\{g\}\}$).

Is there a standard name for this general pattern of composition?

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It's almost conjugation. And can probably be interpreted as such with minor adjustments to the setting. –  Hagen von Eitzen Sep 28 '12 at 19:18
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