Lecture 19: Tempered Distributions

tags: 224a

Distributions are objects which are more general than functions. There are at least three motivations for studying them in the context of mathematical physics:

1. We would like objects which model "point" sources and "roughl" forcing functions.
2. We would like to be able to differentiate whatever these objects are, so we can put them in differential equations.
3. We would like to be able to take the Fourier transform of them.

We will begin by studying tempered distributions, which are a particular subclass of all distributions.

We adopt the perspective that a function

$f(x)$ takes a number

$x$ as an input and ouputs another number. We can think of

$x$ as "probing" or "testing" the function. The key idea is that being able to test each point is unforgiving and leads to "pathological" functions, and we can have a much better behaved theory if we restrict the class of tests.

A tempered distribution takes a Schwartz function as input and outputs a number. Formally, it is a continuous linear functional on the space of Schwartz functions.

$$
The key facts allowing us to extend continuous operators on

$\mathcal{S}(R)$ to

${\mathcal{S}}^{\prime }(R)$ are:

1. The map
   $g\mapsto T_g$ where
   $$T_g\varphi :=\int g(x)\varphi (x)dx$$ is multiplication is easy to see that this map is continuous and 1-1. This yields an embedding of
   $\mathcal{S}(R)$ into
   ${\mathcal{S}}^{\prime }(R)$
2. The embedding is dense, though we will not prove this (the interested reader can see the appendix of RS V.3). This implies that every continuous operator
   $A:\mathcal{S}(R)\to \mathcal{S}(R)$ has a unique extension
   $\hat{A}$ to
   ${\mathcal{S}}^{\prime }(R)$.
3. The extension
   $\hat{A}$ must satisfy (in order to be an extension)
   $$(\hat{A}{T}_g)(\varphi )=T_{Ag}(\varphi )=\int (Ag)(x)\varphi (x)dx\mathrm{\forall }g\in \mathcal{S}(R).$$
   This relation can then be used to derive the definition for any
   $T\in {\mathcal{S}}^{\prime }(R)$, as well as a rule for computing the extension.

We worked this out for the cases

$A=$ multiplication by a slowly growing function or differentiation. The formal development almost exactly follows Reed and Simon V.3, with a couple of extra examples thrown in, so I will not rewrite everything and refer the reader to that section. Ignore any mentions of Frechet spaces and Topology if those things make you uncomfortable.