What does it mean to extend a function by zero outside of its domain?

Question

I have a function $f:\Omega\to \mathbb{R}^n$ with compact support in a domain $\Omega \subseteq \mathbb{R}^2$.. I don't know what it means to extend $f$ by zero outside $\Omega$, I've never heard of that before. If somebody could tell me what this means then that would be much appreciated.

Answer 1

After having spent some time searching, I cannot find an obvious dupe target for this question. I am, frankly, quite surprised by this. However, it is not an unreasonable question, and I think it deserves to be answered on MSE. I am going to answer a slightly more general question:

What does it mean to "extend a function"?

Suppose that $\Omega \subsetneq X$ and that a function $$ f : \Omega \to Y $$ is given. That is, suppose that we are given a function which is defined only on a subset of a smaller set. Given such a function, it is reasonable to ask if there is a natural way of defining $f$ so that it "makes sense" on all of $X$, instead of just a subset. If there is a way of doing this, the result is called an "extension" of $f$. Somewhat more formally:

Definition: Let $\Omega \subsetneq X$ and suppose that $f : \Omega \to Y$. An extension of $f$ is a function $\tilde{f} : X \to Y$ such that $$ \tilde{f}(x) = f(x) \quad\forall x\in\Omega. $$

In other words, $\tilde{f}$ (the extension of $f$) is a new function defined on a larger set than $f$, but which agrees with $f$ on the set where where both functions are defined. Another way of saying this is that the restricttion of $\tilde{f}$ to $\Omega$ is $f$: $$ \tilde{f}|_{\Omega} = f. $$ We can then "extend a function by [blank]", where [blank] describes some property which we want $\tilde{f}$ to have. So, to give a couple of examples:

• To extend $f$ by zero, we define $\tilde{f}$ to be zero away from $\Omega$. That is, we define $$ \tilde{f} : X \to Y : x \mapsto \begin{cases} f(x) & \text{if $x\in \Omega$, and} \\ 0 & \text{otherwise}. \end{cases} $$ In other words, when we extend a function by zero, we simply define the extension to be zero away from the set of interest.

• To extend $f$ by continuity, we define $\tilde{f}$ so that it is continuous on $X$. For this to work, we must assume that $X$ and $Y$ are topological spaces, and, as it wouldn't really make sense to extend a discontinuous function by continuity, we typically also assume that $f$ is continuous on $\Omega$. An elementary example of this might be exponential function: for a fixed (positive) constant it is not too difficult to define $$ \exp_c : \mathbb{Q} \to \mathbb{R} : \frac{p}{q} \mapsto \sqrt[q]{c^p}, $$ where $$ x^p := \prod_{j=1}^{p} x \qquad\text{and}\qquad \left(\sqrt[q]{x}\right)^q = x. $$ This function can be extended by continuity to a function on $\mathbb{R}$ in order to get the usual notion of an exponential function on $\mathbb{R}$.

• To extend $f$ analytically, we define $\tilde{f}$ so that it is analytic on $X$. This notion comes up a lot in complex analysis: if $\Omega \subsetneq \mathbb{C}$ and $f : \Omega \to \mathbb{C}$ is an analytic function, then we might want to know if it is possible to extend $f$ to a function which is analytic on a larger domain. The Riemann zeta function is an example of such a beast: for real $s > 1$, we can define $$ \zeta(s) := \sum_{j=1}^{\infty} j^{-s}. $$ This function can then be extended to a function which is analytic on $\mathbb{C}\setminus\{1\}$. The properties of this extension are of tremendous interest in number theory. :)