Fourier Basis Functions

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Section 4.3 Fourier Basis Functions

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

Activity 16. 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 44. An applet that allows you to explore the shapes of the Fourier basis functions.

Answer.

You should have noticed the following features:

All of the functions are sinusoidal, that is, $\cos(m x)$ or $\sin(m x)$ for integer $m\text{.}$
The coefficient of the constant term ($\cos(m x)$ 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 $\cos(m x)$ or $\sin(m x)\text{.}$

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