What does "test function" mean?

I am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called test functions and why the functions with these properties are needed. I have some idea about these but couldn't understand them properly.

Also, I'll be happy if any one can suggest some good reference on this topic and Sobolev spaces.

Solution 1:

Suppose you want to find the solution of a differential equation, $f'' = gf$ for example.

Take any solution $f$ of this equation, then if you take any function $\psi \in \rm C^{\infty}_c$ it is true to write $$f'' = gf \Longrightarrow \psi f'' = \psi gf \Longrightarrow \int \psi f'' = \int \psi g f \Longrightarrow \int \psi'' f = \int \psi g f$$

Conclusion : any solution of the differential equation $f$ satisfies $\int \psi'' f = \int \psi g f$ but it is possible that functions that are only continuous satisfies the same equation ! These solutions will be called weak solutions because they are solutions of a weaker problem.

Now, why have we chosen the functions $\psi$ to be in $\rm C^{\infty}_c$ ? It is because we transfered the derivation operation from $f$ to $\psi$ by integration by part ; this integration by part goes well only if you suppose that $\psi$ has compact support. I encourage you to do the details of the last implication and it will become clearer.

Solution 2:

To understand why we are calling them test functions, we have to understand what distributions are and where they come from.

Usually, to evaluate a function, we compute its value at the point where we want to know it. But remember that there are spaces of functions (or equivalence classes of functions) such as $L^p$ spaces where the value on a point is not a good representation of the underlying function (it may even have no sense at all). Then, why not trying to evaluate the function as some sort of a weighted mean ? And an integral of $f$ times a function wisely chosen can be seen as a weighted mean.

We define :

$$T_f(\phi)=\int_{\mathbb{R}}f(x) \phi(x) dx.$$

In fact, test functions are just a new way to "know" a function. Because we have the choice of the space from which we take our tests functions, why not taking one with very good properties ? So we do the choice of $\mathbb{C}^{\infty}_{c}$. By doing so, we can proceed to integrate a large variety of functions and we also have linearity and continuity (compared to the topology of $\mathbb{C}^{\infty}_{c}$) of the mapping $T_f$ when it has a meaning. We can also define a lot of useful operations by reporting every problem one would meet with $f$ on the tests functions.

We just have contructed a continuous linear application from $\mathbb{C}^{\infty}_{c}$ to $\mathbb{R}$. Those kind of applications are known as linear forms. So we define that a distribution is a continuous linear form over $\mathbb{C}^{\infty}_{c}$ to $\mathbb{R}$.

This way, we can take $\phi$ being narrower around the point we are considering with constant aera to know the "value" of $f$ near it.

Note that this is just the tip of the icerberg, and that the OP may have understood why we are calling them this way by now. But I wanted to share this POV on this site.

Solution 3:

Specifically why it is called test function. (Essentially not different from another answer.)

Strichartz liked to introduce the concept with physical intuition.

https://books.google.com/books?id=T7vEOGGDCh4C&lpg=PP1&dq=strichartz%20distribution&pg=PA1#v=onepage&q=temperature&f=false

Suppose you want to measure the temperature at a point in the room. You hold out your thermometer, but the tip of it is actually a blob of glass with some liquid inside, so are you really measuring the temperature at any actual point? At best it is a weighted average in a neighborhood of the point.