• Title: Orthogonal Basis

• Series: Fourier Transform

• Chapter: Fourier Series

• YouTube-Title: Fourier Transform 3 | Orthogonal Basis

• Bright video: https://youtu.be/MNVfBEtlwt0

• Dark video: https://youtu.be/XiWAuSBXetk

• Subtitle on GitHub: ft03_sub_eng.srt missing

• 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

• Subtitle in English (n/a)
• 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.

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?