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

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.