Categories: Mathematics.

Schwartz distribution

A Schwartz distribution, also known as a generalized function, is a generalization of a function, allowing us to work with otherwise pathological definitions.

Notable examples of such distributions are the Dirac delta function and the Heaviside step function, whose unusual properties are justified by this generalization.

We define the Schwartz space $\mathcal{S}$ of functions, whose members are often called test functions. Every such $\phi(x) \in \mathcal{S}$ must satisfy the following constraint for any $p, q \in \mathbb{N}$ :

\begin{aligned} \mathrm{max} \big| x^p \phi^{(q)}(x) \big| < \infty \end{aligned}

In other words, a test function and its derivatives decay faster than any polynomial. Furthermore, all test functions must be infinitely differentiable. These are quite strict requirements.

The space of distributions $\mathcal{S}'$ (note the prime) is then said to consist of functionals $\delta[\phi]$ that map a test function $\phi$ from $\mathcal{S}$ to a number from $\mathbb{C}$ ; this is often written as $\Inprod{\delta}{\phi}$ . This notation looks like the inner product of a Hilbert space, and there is a good reason for that: any well-behaved function $f(x)$ can be embedded into $\mathcal{S}'$ by defining the corresponding functional $f[\phi]$ as follows:

\begin{aligned} f[\phi] = \Inprod{f}{\phi} = \int_{-\infty}^\infty f(x) \: \phi(x) \dd{x} \end{aligned}

Not all functionals qualify for $\mathcal{S}'$ : they also need to be linear in $\phi$ , and continuous, which in this context means: if a series $\phi_n$ converges to $\phi$ , then $\Inprod{\delta}{\phi_n}$ converges to $\Inprod{\delta}{\phi}$ for all $\delta$ .

The power of the generalization is that $f(x)$ in fact does not need to be well-behaved: for example, the Dirac delta function can also be used, whose definition is nonsensical outside of an integral, but perfectly reasonable inside one. By treating it as a distribution, we gain the ability to sanely define e.g. its derivatives.

Using the example of embedding a well-behaved function $f(x)$ into $\mathcal{S}$ , we can work out what the derivative of a distribution is using partial integration:

\begin{aligned} \Inprod{f'}{\phi} = \int_{-\infty}^\infty f'(x) \: \phi(x) \dd{x} = \Big[ f(x) \: \phi(x) \Big]_{-\infty}^\infty - \int_{-\infty}^\infty f(x) \: \phi'(x) \dd{x} \end{aligned}

The test function removes the first term, yielding the result $- \Inprod{f}{\phi'}$ . Although this was an example for a well-behaved $f(x)$ , we use it to define the derivative of any distribution $\delta \in \mathcal{S}'$ :

\begin{aligned} \boxed{ \Inprod{\delta'}{\phi} \equiv - \Inprod{\delta}{\phi'} } \end{aligned}

Using the same trick, we can find the Fourier transform (FT) of a distribution. Let us define the FT like so, but be prepared for some switching of the names $k$ and $x$ :

\begin{aligned} \tilde{\phi}(x) = \int_{-\infty}^\infty \phi(k) \exp(- i k x) \dd{k} \end{aligned}

The FT of a well-behaved $f(x)$ is then as follows (this is basically Parseval’s theorem):

\begin{aligned} \inprod{\tilde{f}}{\phi} &= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k} \\ &= \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k} \\ &= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x} \\ &= \inprod{f}{\tilde{\phi}} \end{aligned}

So we defined the FT and derived one of its properties, but to generalize to Schwartz distributions $\delta$ , we swap the roles: the property is an axiom, and serves as the definition of the FT $\tilde{\delta}$ :

\begin{aligned} \boxed{ \inprod{\tilde{\delta}}{\phi} \equiv \inprod{\delta}{\tilde{\phi}} } \end{aligned}

It can be shown that this generalization is a 1:1 mapping between distributions in $\mathcal{S}'$ .

References

K.W. Jacobsen, Note on generalized functions (distributions), 2020, unpublished.