# Information about Fourier Transform - Part 1

• Title: Introduction

• Series: Fourier Transform

• Chapter: Fourier Series

• YouTube-Title: Fourier Transform 1 | Introduction

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

• Dark video: https://youtu.be/XrRxFJIkY1U

• Subtitle on GitHub: ft01_sub_eng.srt missing

• Other languages: German version

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