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Title: Introduction
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 1 | Introduction
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Bright video: https://youtu.be/D_fE_mcByRE
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Dark video: https://youtu.be/XrRxFJIkY1U
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Ad-free video: Watch Vimeo video
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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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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ft01_sub_eng.srt missing
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Other languages: German version
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$. What is not a correct formulation for the Cauchy-Schwarz inequality?
A1: $\langle x, y \rangle \leq \langle x, x \rangle \langle y, y \rangle $
A2: $|\langle x, y \rangle|^2 \leq \langle x, x \rangle \langle y, y \rangle $
A3: $|\langle x, y \rangle| \leq | x | | y| $
A4: $\frac{\langle x, y \rangle}{| x | | y|} \in [-1,1] $ for $x \neq 0 \neq y$.
Q2: Fourier series will be defined for periodic functions. For example, a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $5$-periodic if the number $5$ is a period, which means $f(x + 5 ) = f(x)$ for all $ x \in \mathbb{R}$. Which of the following functions is not $5$-periodic?
A1: $ f(x) = \sin(5 x)$
A2: $ f(x) = 5$
A3: $f(x) = \sin( 2\pi x) $
A4: $f(x) = \cos( \frac{2\pi}{5} x) $
Q3: Let $\langle \cdot, \cdot \rangle$ be the inner product on $\mathcal{P}([-1,1], \mathbb{R})$ given by $\langle f, g\rangle = \int_{-1}^1 f(x) g(x), dx$. Which polynomials are orthogonal to each other? Note that $f,g$ are called orthogonal if $\langle f, g\rangle =0$.
A1: $f(x) = x^2$ and $g(x) = x$
A2: $f(x) = x^2$ and $g(x) = 1$
A3: $f(x) = x$ and $g(x) = x$
A4: $f(x) = x^2$ and $g(x) = x^2$
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Last update: 2024-11
