Fourier Basis Functions

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Section 17.2 Fourier Basis Functions

Before we jump into the details of Fourier series, use the applet below to remind yourself of how the graphs of $\sin\left(\frac{2\pi m x}{L}\right)$ and $\cos\left(\frac{2\pi m x}{L}\right)$ depend on the parameter $m\text{.}$

Activity 17.1. Visualizing the Fourier Basis Functions.

Use the applet below to explore the basis functions. One at a time, set each slider to $1\text{,}$ look at the resulting function, and return that slider to $0\text{.}$ Make a note of any patterns that you see.

Figure 17.2. An applet for manipulating the individual Fourier coefficients (with $L=1$).

Answer.

You should have noticed the following features:

All of the functions are sinusoidal, that is, $\cos(m\pi x)$ or $\sin(m\pi x)$ for integer $m\text{.}$
The coefficient of the constant term ($\cos(m\pi x)$ for $m=0$) is $\frac12 a_0\text{.}$
The coefficients $a_m$ correspond to cosine functions.
The coefficients $b_m$ correspond to sine functions.
The subscript $m$ on the coefficients $a_m$ and $b_m$ corresponds to the value $m$ in $\cos(m\pi x)$ or $\sin(m\pi x)\text{.}$

The basic idea of Fourier series is that any periodic function with period $L$ can be built up as a linear combination of the functions $\cos\left(\frac{2\pi m x}{L}\right)$ or $\sin\left(\frac{2\pi m x}{L}\right)\text{.}$ For any given function, the linear combination is unique.

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