48 lines
1.6 KiB
TeX
48 lines
1.6 KiB
TeX
% !TeX root = ../../script.tex
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\documentclass[../../script.tex]{subfiles}
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\begin{document}
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\section{Outlook: Tempered Distributions}
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\begin{defi}
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A tempered distribution $f$ is a continuous, linear mapping
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\begin{align*}
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f: \mathcal{S}(\realn^d) &\longrightarrow \cmpln \\
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\phi &\longmapsto f(\phi) = (f, \phi) \left( = \int f(x) \phi(x) \dd{x} \right)
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\end{align*}
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\end{defi}
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\begin{thm}
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Tempered distributions are linear, continuous mappings.
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\end{thm}
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\begin{proof}
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To prove linearity, let $\phi, \psi \in \mathcal{S}(\realn^d)$ and $\lambda \in \cmpln$. Then
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\begin{equation}
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(f, \phi + \lambda\psi) = (f, \phi) + \lambda(f, \psi)
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\end{equation}
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For the continuity, we want to consider any sequence $\anyseqdef[\phi]{\mathcal{S}(\realn^d)}$ that converges to $\phi \in \mathcal{S}(\realn^d)$. I.e.
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\begin{equation}
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\lim_{n \rightarrow \infty} \sup_{x \in \realn^d} \abs{x^{\beta} \partial^{\alpha} (\phi_n(x) - \phi(x))} = 0, \quad \forall \alpha, \beta \in \natn_0^d
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\end{equation}
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Then we can conclude that
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\begin{equation}
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\lim_{n \rightarrow \infty} \abs{(f, \phi_n) - (f, \phi)} = 0
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\end{equation}
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\end{proof}
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\begin{rem}
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The space of all tempered distributions is denoted as $\mathcal{S}'(\realn^d)$.
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\end{rem}
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\begin{eg}
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One important example is the Dirac deltra distribution:
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\[
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\delta: \mathcal{S}(\realn^d) \longrightarrow \cmpln
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\]
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It maps a function to its value at $0$.
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\[
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(\delta, \phi) = \int \delta(x) \phi(x) \dd{x} = \phi(0) \in \cmpln
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\]
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\end{eg}
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\end{document} |