• Title: Convergence of Test Functions

• Series: Distributions

• YouTube-Title: Distributions 3 | Convergence of Test Functions

• Bright video: https://youtu.be/HopDSb2ulc4

• Dark video: https://youtu.be/rhHa6zwrP-I

• Subtitle on GitHub: dt03_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

Q1: Let us consider the set of test functions $\mathcal{D}(\mathbb{R}^n) = C_c^\infty( \mathbb{R}^n)$. What do we mean when we write $$\varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi$$

A1: The sequence of functions is uniformly convergent.

A2: There is bounded set such that all supports of $\varphi_k$ lie inside it and the sequence of functions is uniformly convergent.

A3: There is bounded set such that all supports of $\varphi_k$ lie inside it and $$D^\alpha(\varphi_k) \rightarrow D^\alpha(\varphi)$$ uniformly for all multi-indices.

Q2: Does the sequence $(\varphi_k)$ with $\varphi_k = 0$ converge in the space $\mathcal{D}(\mathbb{R}^n)$?

A1: Yes.

A2: No.

Q3: Consider the function $\varphi_k \in \mathcal{D}(\mathbb{R}^n)$ given by $$\varphi_k(x) = \begin{cases} 0 &,~ | x | \geq 1 \ \frac{1}{k} \exp\left( - \frac{1}{1 - |x |^2 } \right) &,~ | x | < 1 \end{cases}$$ Which claim is true?

A1: $$\varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} 0$$

A2: $$\varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi_1$$

A3: $$\varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} 1$$

A4: $$\varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi_2$$

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