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Title: Pointwise Convergence of Fourier Series
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 17 | Pointwise Convergence of Fourier Series
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Subtitle on GitHub: ft17_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: \mathbb{R} \rightarrow \mathbb{C}$ a $C^1$-function that is also $2\pi$-periodic. What is not correct for the Fourier series $\mathcal{F}_n(f)$ in general?
A1: $\mathcal{F}_n(f) \xrightarrow{n\to \infty} f$ pointwisely.
A2: $\mathcal{F}_n(f) \xrightarrow{n\to \infty} f$ in the $L^2$-norm.
A3: $\mathcal{F}_n(f) \xrightarrow{n\to \infty} f$ uniformly.
A4: The pointwise limit $\lim_{n \rightarrow \infty}\mathcal{F}_n(f)$ is not continuous.
Q2: Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a continuous function that is also $2\pi$-periodic. Do we have $\mathcal{F}_n(x) \xrightarrow{n \to \infty} f(x)$ for every $x \in [-\pi, \pi]$?
A1: No, in general we don’t have the pointwise convergence.
A2: Yes, we have this for every point where the function is continuous.
Q3: Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be the function $x \mapsto 2x$ on the inverval $[-\pi, \pi)$ and let’s extend it $2\pi$-periodically. What is the value of $\mathcal{F}_n(\pi)$?
A1: $0$
A2: $1$
A3: $2$
A4: $\pi$
A5: $-\pi$
A6: $\frac{1}{2}$
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Date of video: 2024-12-06
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Last update: 2025-09
