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There exists notation for the pointwise multiplication of a function $j$ or $k$. This is often denoted as $j\cdot k$ or $j \circ k$ using the Hadamard notation. Consider the pointwise exponentiation of some function $f:X \rightarrow Y$ with exponent $n$ denoted by $g:Y\rightarrow Z$:

$$Z = \left\{x_i \in X|f(x_i)^{n}\right\}$$

How can the operation of pointwise exponentiation be notated? Does $g = f^n$ suffice? For instance in the programming language Matlab this operation would be expressed as X.^n to distinguish it from X^n which denotes matrix multiplication.

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For positive $n$, the notation $f^n$ usually means repeated composition as per Wikipedia: Exponential Notation for Function Names. An exception to this rule is trigonometric and logarithmic functions where the exponent always means repeated multiplication. Thus

$$ \sin^2 x = (\sin x) (\sin x) $$ $$ \log^2 x = (\log x) (\log x) $$

I think it is ok to use $f^n$ to denote point wise exponentiation (repeated product)

$$ f^n(x) = \big(f(x) \big)^n $$

as long as you clearly define it.

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