-
Title: Convolution
-
Series: Distributions
-
Chapter: Operations on Distributions
-
YouTube-Title: Distributions 13 | Convolution
-
Bright video: Watch on YouTube
-
Dark video: Watch on YouTube
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: dt13_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Related videos:
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: For two integrable functions $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$, one can define the convolution. What is the correct definition?
A1: $ \displaystyle (f \ast g)(x) $ $ \displaystyle = \int_{\mathbb{R}^n} f(x-y) g(y) , dy $
A2: $ \displaystyle (f \ast g)(x) $ $ \displaystyle = \int_{\mathbb{R}^n} f(x) g(x-y) , dy $
A3: $ \displaystyle (f \ast g)(x) $ $ \displaystyle = \int_{\mathbb{R}^n} f(y) g(y) , dy $
A4: $ \displaystyle (f \ast g)(x) $ $ \displaystyle = \int_{\mathbb{R}^n} f(y) g(x-y) , dx $
Q2: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a locally integrable function and $\varphi, \psi \in \mathcal{D}(\mathbb{R}^n)$. Define $\check{\psi}(z) := \psi(-z)$. What is correct?
A1: $ \displaystyle \int_{\mathbb{R}^n} (\psi \ast f)(x) \varphi(x) , dx $ $ \displaystyle = \int_{\mathbb{R}^n} f(x) (\check{\psi} \ast \varphi)(x) , dx$
A2: $ \displaystyle \int_{\mathbb{R}^n} (\psi \ast f)(x) \varphi(x) , dx $ $ \displaystyle = \int_{\mathbb{R}^n} f(x) (\psi \ast \varphi)(x) , dx$
A3: $ \displaystyle \int_{\mathbb{R}^n} (\psi \ast f)(x) \varphi(x) , dx $ $ \displaystyle = \int_{\mathbb{R}^n} f(x) (\varphi \ast \psi)(x) , dx$
A4: $ \displaystyle \int_{\mathbb{R}^n} (\psi \ast f)(x) \varphi(x) , dx $ $ \displaystyle = \int_{\mathbb{R}^n} \psi(x) (\check{f} \ast \check{\varphi})(x) , dx$
Q3: Let $T$ be a distribution and $\psi$ a test function. What is the correct definition for the convolution $\psi \ast T$?
A1: $ \displaystyle (\psi \ast T) ( \varphi) = T(\check{\psi} \ast \varphi) $
A2: $ \displaystyle (\psi \ast T) ( \varphi) = T(\psi \ast \varphi) $
A3: $ \displaystyle (\psi \ast T) ( \varphi) = T(\psi \cdot \varphi) $
A4: $ \displaystyle (\psi \ast T) ( \varphi) = T(\check{\psi} \cdot \varphi) $
-
Last update: 2024-11
