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I am a lowly data analyst, but I like to use standard mathematical terms and notation when possible. Here is the setting: given some function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$, and some subset $S \subset \mathbb{R}^2$, I define "zero-extended restrictions" of $f$ for each of the two disjoint regions into which I have divided the plane:

$$ f_1 : \mathbb{R}^2 \rightarrow \mathbb{R}, \quad f_1(x) = \begin{cases} f(x), &x \in S \\ 0, &x \notin S \end{cases} $$

and

$$ f_2 : \mathbb{R}^2 \rightarrow \mathbb{R}, \quad f_2(x) = \begin{cases}0, &x \in S \\ f(x), &x \notin S \end{cases} $$

$f_1$ and $f_2$ are like restrictions, but not exactly, because I don't want to change the domain, which is still all of $\mathbb{R}^2$. Crucially, I need equations like $f(x) = f_1(x) + f_2(x)$ to still make sense. Therefore it seems like the restriction notation $f|_S$ would be wrong, i.e. $f_1 \neq f|_S$.

Is there some other concise term or notation to describe these functions $f_1$ and $f_2$? Or are they sufficiently unusual creations that I won't make a fool of myself if I coin my own description and notation?

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2 Answers 2

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In terms of the characteristic function of a set $S$, defined by $$ \chi_{S}(x) = \begin{cases} 1 & x \in S, \\ 0 & x \not\in S, \end{cases} $$ we have $f_{1} = f\cdot\chi_{S}$. Writing $S'$ for the complement of $S$, we also have $f_{2} = f\cdot\chi_{S'} = f\cdot(1 - \chi_{S})$.

It's not unreasonable to introduce special notation, e.g., $f_{1} = f_{S}$, if $f\cdot\chi_{S}$ is used a lot.

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We can also use Iverson brackets to define $f_1$ and $f_2$ from $f$. We set \begin{align*} f_1&:= f[x\in S]\\ f_2&:=f[x\notin S] \end{align*} and obtain this way \begin{align*} \color{blue}{f=f_1+f_2} \end{align*}

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