Chapter 16 Distribution theory
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Example 16.1. Consider the ODE initial-value problem
\[ u'(x)=H(x),\qquad u(0)=0, \]
where
\[ H(x)=\begin {cases} 0&x\leq 0\\ 1&x>0. \end {cases} \]
It seems clear that
\[ u(x)=\begin {cases} 0&x\leq 0\\ x&x>0, \end {cases} \]
should be the solution to this. However, this \(u\) is not differentiable at zero, so strictly speaking it does not satisfy \(u'(x)=H(x)\) at \(x=0\).
The space of locally integrable functions \(L^1_{\rm loc}(\mR )\) is defined as consisting of those functions \(f:\mR \to \mC \) for which \(\int _a^b |f(x)|\,dx<\infty \) for all \(a,b\in \mR \). For example all continuous functions are locally integrable.
The space of infinitely differentiable functions with compact support \(\mathcal {D}(\mR )\) consists of those functions \(\varphi :\mR \to \mR \) which are infinitely differentiable on \(\mR \) and are such that there exist \(a,b\in \mR \) such that \(\varphi (x)=0\) for all \(x\notin [a,b]\).
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Definition 16.2. A distribution \(T\) is a continuous linear operator \(T:\mathcal {D}(\mR )\to \mC \). The regular distribution \(T_f\) corresponding to \(f\in L^1_{\rm loc}(\mR )\) is the distribution
\[ \varphi \mapsto \int _\mR f(x)\varphi (x)\,dx. \]
We denote the space of distributions by \(\mathcal {D}'(\mR )\).
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Remark 16.3. We have been deliberately vague about the sense of continuity in Definition 16.2 since it is complicated, but will never be an issue. We note that the topology on \(\mathcal {D}(\mR )\) which is needed for this does not come from a metric, so the more general notions of MA30055 Introduction to Topology would be needed.
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Remark 16.5. Let \(\eta \in L^1(\mR )\) be such that \(\int _\mR \eta (x)\,dx=1\). An example of such a function is \(\eta (x)=\frac {1}{\sqrt {2\pi }}\e ^{-x^2/2}\). For \(\varepsilon >0\) define \(\eta _\varepsilon (x)=\varepsilon ^{-1}\eta (\varepsilon ^{-1}x)\) (the idea is that as \(\varepsilon \) gets smaller, \(\eta _\varepsilon \) gets “narrower” and “higher” whilst always having integral 1). Then for the distribution corresponding to the function \(\eta _\varepsilon \) we have \(\lim _{\varepsilon \downarrow 0}T_{\eta _\varepsilon }=\delta \). Note that (with \(y=\varepsilon ^{-1}x\)) formally
\[ T_{\eta _\varepsilon }(\varphi )=\int _\mR \varepsilon ^{-1}\eta (\varepsilon ^{-1}x)\varphi (x)\,dx =\int _\mR \eta (y)\varphi (\varepsilon y)\,dy\to \varphi (0)\int _\mR \eta (y)\,dy=\varphi (0)=\delta (\varphi ). \]
If \(f\in L^1_{\rm loc}(\mR )\) is such that \(f'\in L^1_{\rm loc}(\mR )\), then we have for \(\varphi \in \mathcal {D}(\mR )\)
\[ \int _\mR f'(x)\varphi (x)\,dx =\left [f(x)\varphi (x)\right ]_{x=-\infty }^\infty -\int _\mR f(x)\varphi '(x)\,dx =-\int _\mR f(x)\varphi '(x)\,dx, \]
where we have used integration by parts and the fact that \(\varphi \) is zero outside of some compact interval. We therefore have
\[ T_{f'}(\varphi )=-T_f(\varphi '). \]
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Example 16.7. The differential equation \(u'=H\) from Example 16.1 should be understood in the sense of distributions, i.e. \(T_{u}'=T_{H}\). We have
\[ T_u'(\varphi ) =-T_u(\varphi ') =-\int _{-\infty }^\infty u(x)\varphi '(x)\,dx =-\int _0^\infty x\varphi '(x)\,dx, \]
and
\[ T_H(\varphi ) =\int _{-\infty }^\infty H(x)\varphi (x)\,dx =\int _0^\infty \varphi (x)\,dx. \]
From integration by parts we indeed have
\[ -\int _0^\infty x\varphi '(x)\,dx = \int _0^\infty \varphi (x)\,dx. \]
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Example 16.8. With \(H\) as in Example 16.1 we have \(T_H'=\delta \). This can be seen as follows:
\[ T_H'(\varphi ) =-T_H(\varphi ') =-\int _{-\infty }^\infty H(x)\varphi '(x)\,dx =-\int _{0}^\infty \varphi '(x)\,dx =\varphi (0) =\delta (\varphi ), \]
which shows that \(T_H'=\delta \).
Note that formally we have
\[ T_{\psi f}(\varphi ) =\int _\mR \left [\psi (x)f(x)\right ]\varphi (x)\,dx =\int _\mR f(x)\left [\psi (x)\varphi (x)\right ]\,dx =T_f(\psi \varphi ). \]
Since for \(\psi \in C^\infty (\mR )\) and \(\varphi \in \mathcal {D}(\mR )\) we have \(\psi \varphi \in \mathcal {D}(\mR )\), we can make the following definition.