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Title: Orthogonal Basis
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 3 | Orthogonal Basis
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Bright video: https://youtu.be/MNVfBEtlwt0
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Dark video: https://youtu.be/XiWAuSBXetk
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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: ft03_sub_eng.srt missing
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Timestamps
00:00 Introduction
00:45 Real trigonometric polynomials
01:30 Subspace of trigonometric polynomials
02:41 Inner product for trigonometric polynomials
04:11 Examples
11:54 Result: OB for trigonometric polynomials
12:55 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: What dimension has the space spanned by the functions $x \mapsto 1$, $x \mapsto \cos(kx)$, $x \mapsto \sin(kx)$ where $k$ goes from $1$ to a fixed $n \in \mathbb{N}$?
A1: $2n + 1$
A2: $n$
A3: $2n$
A4: $\infty$
Q2: Consider the inner product $\langle f, g\rangle = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) g(x) , dx$. Are the two functions $x \mapsto \cos(2x)$ and $x \mapsto \cos(4x)$ orthogonal?
A1: Yes, the inner product is zero.
A2: No, the inner product is not zero.
A3: No, the inner product cannot be calculated.
A4: One needs more information.
Q3: Consider the space spanned by the functions $x \mapsto 1$, $x \mapsto \cos(kx)$, $x \mapsto \sin(kx)$ where $k$ goes from $1$ to a fixed $n \in \mathbb{N}$ and also consider the inner product $\langle f, g\rangle = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) g(x) , dx$ Do these elements form an orthonormal basis?
A1: No, just an orthogonal basis.
A2: Yes, they do.
A3: No, they don’t have the orthogonality.
A4: No, they are not even a basis.
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Last update: 2024-11
