TITLE: Terminology and notation for a zero-padded restriction of a function
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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*}