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Title: Fundamental Example for Fourier Series
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 10 | Fundamental Example for Fourier Series
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Python file: Download Python file
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ft10_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps
00:00 Introduction
00:47 Parseval’s identity for step functions
01:47 Fundamental Step Function
03:42 Fourier Series for Step Function
05:53 Visualization
08:10 Proof of Parseval’s identity
15:37 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: What is $c_0$ for the function $h_a$ that is $1$ on the interval $[-\pi, a]$ and $0$ on the interval $(0,\pi)$?
A1: $c_0 = \frac{a + \pi}{2 \pi}$
A2: $c_0 = \frac{a - \pi}{2 \pi}$
A3: $c_0 = \frac{a}{2 \pi}$
A4: $c_0 = \frac{\pi - a}{2 \pi}$
Q2: Having $c_k$ for all $k \neq 0$. How do you get the real coefficients $a_k$ and $b_k$?
A1: $a_k = 2 , \mathrm{Re}( c_k )$ and $b_k = - 2 ,\mathrm{Im}( c_k )$
A2: $a_k = 2 ,\mathrm{Im}( c_k )$ and $b_k = - 2, \mathrm{Re}( c_k )$
A3: $a_k = 2, \mathrm{Re}( c_k )$ and $b_k = 2 ,\mathrm{Im}( c_k )$
A4: $a_k = 2 ,\mathrm{Im}( c_k )$ and $b_k = 2 ,\mathrm{Re}( c_k )$
Q3: We know that Parseval’s identity holds for the function $h_a$ that is $1$ on the interval $[-\pi, a]$ and $0$ on the interval $(0,\pi)$. What does it mean?
A1: $ \sum_{ k = -\infty}^{\infty} |c_k|^2 = | h_a |^2 $
A2: $ \sum_{ k = 1}^{\infty} |a_k|^2 + \sum_{ k = 1}^{\infty} |b_k|^2 = | h_a |^2 $
A3: $ \sum_{ k = 0}^{\infty} |a_k|^2 = | h_a |^2 $
A4: $ \sum_{ k = -\infty}^{\infty} |c_k| = | h_a | $
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Last update: 2025-09
