The next basic tool we’re going to need is the theory of distributions.

Distributions

Distributions are extremely useful because they are both fairly general (including both all integrable functions and things like the Dirac delta function) but also allow for operations such as differentiation. So oftentimes we can obtain distribution solutions to differential equations we are interested in.

Actually, we’ll only discuss here tempered distributions. A tempered distribution is a linear functional ${\phi: \mathcal{S} \rightarrow \mathbb{C}}$. Clearly the tempered distributions form a vector space ${\mathcal{S}'}$; it is a locally convex space if we endow it with the weak* topology. It must now be seen how distributions generalize functions. So, if ${f \in L^p(\mathbb{R}^n)}$ for any ${p, 1 \leq p \leq \infty}$, then ${f}$ can be made into a distribution

$\displaystyle w \rightarrow \int_{\mathbb{R}^n} wf dx, \ w \in \mathcal{S}.$

In fact, we could just assume that ${f}$ grows polynomially. In particular, we have an imbedding ${\mathcal{S} \rightarrow S'}$.

An example of a distribution that is not a function is the Dirac distribution ${\delta}$ mapping ${f \rightarrow f(0)}$. (more…)

So, as I’ve already indicated, I’m planning to talk about PDEs for the next month or so, both the general theory and specific posts on the equations of mathematical physics.  There are some preliminaries I’ll have to do first such as Fourier transforms.  Today, I’ll get up to the inversion formula for Schwarz functions.

The Schwarz Class

The Schwarz class ${\mathcal{S}}$ consists of smooth functions ${f: \mathbb{R}^n \rightarrow \mathbb{R}}$ such that for all multi-indices ${\alpha=(\alpha_1, \dots, \alpha_n), \beta=(\beta_1, \dots, \beta_n)}$,

$\displaystyle x^{\alpha} D^{\beta} f := x_1^{\alpha_1} \dots x_n^{\alpha_n} \left( \frac{\partial}{\partial x_1}\right)^{\beta_1} \dots \left( \frac{\partial}{\partial x_n}\right)^{\beta_n}f$

is bounded. For instance, the function ${e^{-|x|^2}}$ is in ${\mathcal{S}}$, as is any ${C^{\infty}}$ function with compact support. Elements in ${\mathcal{S}}$ are loosely speaking, functions that decrease rapidly at ${\infty}$ with all their partial derivatives.

There is a way to make the space ${\mathcal{S}}$ into a Frechet space, by the countable family of seminorms

$\displaystyle ||f||_{a,b} := \sup |x^{a} D^{b} f(x) |.$ (more…)