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Title: Finite-Order Distributions
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Series: Distributions
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YouTube-Title: Distributions 12 | Finite-Order Distributions
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Bright video: Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: dt12_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Definitions in the video: distribution of finite order
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $T$ be a distribution of finite order $0$. What is correct?
A1: For every compact set $K \subseteq \mathbb{R}^n$, we find a test function $\varphi$ with support in $K$ such that $$ | \langle T, \varphi \rangle | \leq | \varphi |_\infty$$
A2: For every compact set $K \subseteq \mathbb{R}^n$, there is a constant $c > 0$ such that every test function $\varphi$ with support in $K$ satisfies $$ | \langle T, \varphi \rangle | \leq c \cdot | \varphi |_\infty$$
A3: For every test function $\varphi$ with support in $K \subseteq \mathbb{R}^n$, there is a constant $c>0$ such that $$ | \langle T, \varphi \rangle | \leq c \cdot | \varphi |_\infty$$
A4: For every test function $\varphi$ and every constant $c>0$, we find a compact set $K \subseteq \mathbb{R}^n$ such that $$ | c \cdot \langle T, \varphi \rangle | \leq | \varphi |_\infty$$
A5: None of the other statements is correct.
Q2: What is not correct for a regular distribution $T_f$?
A1: $T_f$ is a distribution of finite order.
A2: $T_f$ is a distribution of order $m = 0$.
A3: $T_f$ is not a distribution of finite order.
A4: $T_f$ can be represented by a complex Radon measure $\mu$.
A5: $T_f$ is a distribution of order $m = 1$.
Q3: Is the delta distribution a distribution of order $0$?
A1: No, it’s not a regular distribution and, therefore, also not of finite order.
A2: No, it’s only of order $1$.
A3: Yes, it’s of finite order $0$.
A4: No, because it cannot be represented by a complex Radon measure $\mu$.
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Last update: 2024-11
