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diff --git a/source/know/concept/schwartz-distribution/index.md b/source/know/concept/schwartz-distribution/index.md index 2492d91..3d5186f 100644 --- a/source/know/concept/schwartz-distribution/index.md +++ b/source/know/concept/schwartz-distribution/index.md @@ -11,7 +11,7 @@ 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 +Notable examples of such distributions are the [Dirac delta function](/know/concept/dirac-delta-function/) and the [Heaviside step function](/know/concept/heaviside-step-function/), whose unusual properties are justified by this generalization. @@ -31,13 +31,13 @@ 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 $$\mathcal{S}$$, +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 $$\Inprod{f}{\phi}$$. +this is often written as $$\Inprod{\delta}{\phi}$$. This notation looks like the inner product of a [Hilbert space](/know/concept/hilbert-space/), -for good reason: any well-behaved function $$f(x)$$ can be embedded +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: $$\begin{aligned} @@ -49,10 +49,11 @@ $$\begin{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 $$\Inprod{f}{\phi_n}$$ -converges to $$\Inprod{f}{\phi}$$ for all $$f$$. +converges to $$\phi$$, then $$\Inprod{\delta}{\phi_n}$$ +converges to $$\Inprod{\delta}{\phi}$$ for all $$\delta$$. -The power of this generalization is that $$f(x)$$ does not need to be well-behaved: +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. @@ -60,7 +61,7 @@ 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: +we can work out what the derivative of a distribution is using partial integration: $$\begin{aligned} \Inprod{f'}{\phi} @@ -68,20 +69,21 @@ $$\begin{aligned} = \Big[ f(x) \: \phi(x) \Big]_{-\infty}^\infty - \int_{-\infty}^\infty f(x) \: \phi'(x) \dd{x} \end{aligned}$$ -The test function removes the boundary term, yielding the result -$$- \Inprod{f}{\phi'}$$. Although this was an example for a specific $$f(x)$$, -we use it to define the derivative of any distribution: +The test function removes the first term, +yielding the result $$- \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{f'}{\phi} = - \Inprod{f}{\phi'} + \Inprod{\delta'}{\phi} + \equiv - \Inprod{\delta}{\phi'} } \end{aligned}$$ Using the same trick, we can find the -[Fourier transform](/know/concept/fourier-transform/) (FT) -of a generalized function. -We define the FT as follows, +[Fourier transform](/know/concept/fourier-transform/) (FT) of a distribution. +Let us define the FT like so, but be prepared for some switching of the names $$k$$ and $$x$$: $$\begin{aligned} @@ -89,30 +91,34 @@ $$\begin{aligned} = \int_{-\infty}^\infty \phi(k) \exp(- i k x) \dd{k} \end{aligned}$$ -The FT of a Schwartz distribution $$f$$ then turns out to be as follows: +The FT of a well-behaved $$f(x)$$ is then as follows +(this is basically [Parseval's theorem](/know/concept/parsevals-theorem/)): $$\begin{aligned} \inprod{\tilde{f}}{\phi} &= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k} - = \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k} + \\ + &= \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k} \\ &= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x} - = \inprod{f}{\tilde{\phi}} + \\ + &= \inprod{f}{\tilde{\phi}} \end{aligned}$$ -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: +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 $$\tilde{\delta}$$: $$\begin{aligned} \boxed{ - \inprod{\tilde{f}}{\phi} - = \inprod{f}{\tilde{\phi}} + \inprod{\tilde{\delta}}{\phi} + \equiv \inprod{\delta}{\tilde{\phi}} } \end{aligned}$$ +It can be shown that this generalization is a 1:1 mapping +between distributions in $$\mathcal{S}'$$. + ## References |