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I have been pondering something lately and asked myself whether there is a mathematical construct that takes a function as its parameter and returns another function as its result?

That is, it is a function of a function that produces a function. Is there a term for this? What are some basic facts and where can I learn more about such structures?

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    $\begingroup$ Yes there is, en.m.wikipedia.org/wiki/List_of_transforms , the most common and used kind of transform is integral transform, en.m.wikipedia.org/wiki/Integral_transform $\endgroup$
    – ℋolo
    Oct 17, 2017 at 3:32
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    $\begingroup$ There is a term: we call such thing functions! $\endgroup$ Oct 17, 2017 at 3:35
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    $\begingroup$ Your functional analysis tag is a good one. Follow that. Also, these things are everywhere. If you think a bit you can create your own pretty easily. For instance, $f \mapsto f(cx)$ is such a transform where $c$ is a fixed constant and $f : \mathbb{R} \to \mathbb{R}$. $\endgroup$
    – Randall
    Oct 17, 2017 at 3:35
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    $\begingroup$ In functional analysis we call those guys operators. $\endgroup$
    – amsmath
    Oct 17, 2017 at 3:56
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    $\begingroup$ It is a quite common situation where functions of interest comprise points of certain structure (which we often call function space) and then we are interested in maps between such structures. Operator theory is one such example. $\endgroup$ Oct 17, 2017 at 3:56

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These are often called operators. Some operators take two arguments, called binary, like $()\times()$ or $()-()$; others take only one, called unary, such as $()^2$.

Some examples of operators or “functions” that take functions $f(x)$ and return functions include

  • the derivative $d/dx$
  • the derivative’s half-brother the gradient $\nabla$
  • the derivative’s twin sister the partial derivative $\partial/\partial x$
  • exponentiation $[f(x)]^2$
  • multiplication $2\times f(x)$
  • addition $f(x)+2$
  • function composition $[g\circ f](x)$ if the domain of $g$ and range of $f$ align nicely
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