• Title: Orthonormalbasis of Trigonometric Functions

• Series: Fourier Transform

• Chapter: Fourier Series

• YouTube-Title: Fourier Transform 4 | Orthonormalbasis of Trigonometric Functions

• Bright video: https://youtu.be/6Z8EXPFH-pk

• Dark video: https://youtu.be/0bhXht8mDUY

• Quiz: Test your knowledge

• Subtitle on GitHub: ft04_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

Q1: Consider the trigonometric polynomial $f(x) = 1 + \sum_{k=1}^n a_k \cos(k x)$. How can the coefficients $a_k$ be calculated?

A1: $a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} \cos(kx) f(x) , dx$

A2: $a_k = \int_{-\pi}^{\pi} \cos(kx) f(x) , dx$

A3: $a_k = \frac{1}{2 \pi} \int_{-\pi}^{\pi} \cos(kx) f(x) , dx$

A4: $a_k = -\frac{1}{2\pi} \int_{-\pi}^{\pi} \cos(kx) f(x) , dx$

Q2: Consider the trigonometric polynomial $f(x) = a_0 + \sum_{k=1}^n a_k \cos(k x)$. How can the coefficient $a_0$ be calculated?

A1: $a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) , dx$

A2: $a_0 = \frac{1}{\sqrt{2}\pi} \int_{-\pi}^{\pi} f(x) , dx$

A3: $a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) , dx$

A4: $a_0 = \frac{\sqrt{2}}{\pi} \int_{-\pi}^{\pi} f(x) , dx$

Q3: Consider the trigonometric polynomial $f(x) = a_0 + \sum_{k=1}^n a_k \cos(k x)+ \sum_{k=1}^n b_k \sin(k x)$. How can the coefficients $b_k$ be calculated?

A1: $b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} \sin(kx) f(x) , dx$

A2: $b_k = \int_{-\pi}^{\pi} \sin(kx) f(x) , dx$

A3: $b_k = \frac{1}{2 \pi} \int_{-\pi}^{\pi} \sin(kx) f(x) , dx$

A4: $b_k = -\frac{1}{2\pi} \int_{-\pi}^{\pi} \sin(kx) f(x) , dx$

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