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 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 \(f[\phi]\) which map a test function \(\phi\) from \(\mathcal{S}\), to a number from \(\mathbb{C}\), which is often written as \(\braket{f}{\phi}\). This notation looks like the inner product of a Hilbert space, for good reason: any well-behaved function \(f(x)\) can be embedded into \(\mathcal{S}'\) by defining the corresponding functional \(f[\phi]\) as follows:

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

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 \(\braket{f}{\phi_n}\) converges to \(\braket{f}{\phi}\) for all \(f\).

The power of this generalization is that \(f(x)\) 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:

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

The test function removes the boundary term, yielding the result \(- \braket{f}{\phi'}\). Although this was an example for a specific \(f(x)\), we use it to define the derivative of any distribution:

\[\begin{aligned} \boxed{ \braket{f'}{\phi} = - \braket{f}{\phi'} } \end{aligned}\]

Using the same trick, we can find the Fourier transform (FT) of a generalized function. We define the FT as follows, 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 Schwartz distribution \(f\) then turns out to be as follows:

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

Note that the ordinary FT \(\tilde{f}(k) = \hat{\mathcal{F}}\{f(x)\}\) is already a 1:1 mapping of test functions \(\phi \leftrightarrow \tilde{\phi}\). As it turns out, in this generalization it is also a 1:1 mapping of distributions in \(\mathcal{S}'\), defined as:

\[\begin{aligned} \boxed{ \braket*{\tilde{f}}{\phi} = \braket*{f}{\tilde{\phi}} } \end{aligned}\]

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