Graph od periodic extension of function and its Fourier cosine series

calculusfourier analysisfourier seriessequences-and-seriestrigonometric series

The function $f$ is defined as follows:

$ f(x)=
\left\{
\begin{array}{ll}
1 & ,x \in ( \frac{3}{2}, 2) \\
3-x & ,x \in [2,3)
\end{array}
\right.
$

I have to find the Fourier cosine series of this function.

  1. Could someone just write or sketch the periodic extension of $f$ on whole real line? What does the even periodic extension look like and what happens on $[0, \frac{3}{2} ]$?

  2. When we get the even periodic extension, how are coefficients $a_0, a_n, b_n $ calculated? I suppose that the period is $3 $ but what are the limits of integrals?

I know how to sketch the extension and find coefficients when the function is given on interval of the form $[0,l]$, but here it is not the case, it starts at $\frac{3}{2}$.
I would appreciate any help.

Best Answer

What's the length of the interval where your function is defined? The domain is $(\frac 32,3)$. The length of this interval is $\frac 32$. So you can just shift everything by that value, and your interval will become $(0,\frac 32)$. Then the problem reduces to something that you already know how to solve: $$ f(x)= \left\{ \begin{array}{ll} 1 & ,x \in ( 0, \frac{1}{2}) \\ 3-x & ,x \in [\frac 12,\frac32) \end{array} \right. $$ While this function is periodic, it is not even. You can extend it as: $$ f(x)= \left\{ \begin{array}{ll} x & ,x \in ( 0, 1) \\ 1 &, x\in [1,2)\\ 3-x & ,x \in [2,3) \end{array} \right. $$ Now you can make an even, periodic function out of it.

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