Understanding Distributions

Question

I have been studying distributions, and I am still trying to get the intuition behind the following:

$1)$ Suppose $f\in L^1_{loc}(\mathbb{R})$, and $\phi\in C^{\infty}_c(\mathbb{R})$. A distribution is defined as $\langle T_f, \phi \rangle=\int_{\mathbb{R}}f\phi dx$.

Suppose we are given an $f$, is the definition telling us we can define a distribution for any $\phi$? In other words, if we have some given $L^1_{loc}$ function, we can define a different distribution for say $\phi_1=sin(x)\chi_{[0,1]}$ and $\phi_2=e^x\chi_{[0,2]}$?

$2)$ From this definition, is it valid to define a function as follows:

$g(x)=\int_{\mathbb{R}}f\phi dx$ with $x\in [x-\epsilon, x+\epsilon]$ and $\phi$ compactly supported on $[x-\epsilon, x+\epsilon]$? where $f$ and $\phi$ are some appropriate functions?

$3)$ If $S(\mathbb{R})$ is the Schwarz space of rapidly decreasing functions, and $S'(\mathbb{R})$ is the dual space (space of tempered distributions), does this just mean that for $f\in S(\mathbb{R})$, the Fourier transform $\hat f$ defines a distribution which takes finite values?

I think I am most stuck trying to make sense of the third one.

Answer 1

1) It would be more correct to write:

Suppose $f \in L^1_{\text{loc}}(\mathbb R)$. Then we define a distribution $T_f$ through $\langle T_f, \phi \rangle = \int f(x) \, \phi(x) \, dx$ for $\phi \in C_c^\infty(\mathbb R).$ The distribution is not defined for one $\phi$. It is defined for all $\phi \in C_c^\infty(\mathbb R).$

Also, as I wrote in a comment: your $\phi_1$ and $\phi_2$ don't belong to $C_c^\infty(\mathbb R).$

2) The expression $\int_{\mathbb{R}} f(x) \, \phi(x) \, dx$ doesn't define a function of $x$ since $x$ is not free in the expression. You could equally well write $g(x) = \int_{\mathbb{R}} f(t) \, \phi(t) \, dt$; where does $x$ come from? The function $g$ would be constant.

3) I don't understand your question; a distribution always take finite values.

Answer 2

For 1) there is not a distribution for every $\phi$. The distribution is the linear functional

$$\phi \longmapsto \int_{\mathbb R} f \phi dx$$

This functional is denoted $T_f$.