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Let $L$ be an operator that maps any well-behaved (with bounded integral) function $f:\mathbb{R} \rightarrow\mathbb{R}$ to a number $L_f\in\mathbb{R}$, such that if $g(x)=f(x-d)$ for all $x\in\mathbb{R}$, then $L_g=L_f+d$.

Subject to the above condition, $L_f$ could be, for example:

  • the centroid $\frac{\int_{-\infty}^\infty xf(x)dx}{\int_{-\infty}^\infty f(x)dx}$
  • $\inf \arg \max_x⁡{f(x)}$
  • the median
  • $\inf \arg \max_x⁡{f'(x)}$

or any other domain point serving as some kind of "anchor" to the function's "signature", such that it "moves along" when the function is translated.

Is there a name for such a property/operator in general?

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  • $\begingroup$ I like this idea but the $\arg\max_x$ examples don't seem to work -- what do you do when the max is not unique? Maybe you could define instead $\inf \arg\max_x f(x)$. $\endgroup$
    – 6005
    Sep 24, 2019 at 11:01
  • $\begingroup$ 6005 yes indeed $\endgroup$
    – Museful
    Sep 24, 2019 at 11:08
  • $\begingroup$ Centroid may fail to exist, such as for the pdf of the Cauchy distribution. $\endgroup$
    – user21820
    Sep 26, 2019 at 13:12

1 Answer 1

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I don't suspect there is a standard term, but I might call such a map $L$ horizontal-translation-preserving because it preserves horizontal translation of the input function by mapping it to translation of the output real number. Formally, $$ L: (\mathbb{R} \to \mathbb{R}) \to \mathbb{R} $$ preserves horizontal translation (also called "shifting") if it satisfies $$ L(x \mapsto f(x - d)) = L(f) + d. $$

(I would prefer just "translation-preserving", but one has to distinguish between translation in the $x$- and $y$-direction of a function.)


Note that such a map $L$ need not be a linear operator (see e.g. the $\inf \arg \max$ example), and it also need not be a bounded linear operator, which are the objects normally studied in functional analysis. So I would be careful when saying "let $L$ be an operator" that you clarify you don't mean linear or bounded linear.

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