Fourier series, estimate the values of $a_0$, $a_n$ and $b_n$.

Question

Using the following periodic function (period of $2\pi$)

$$F (x) =\begin {cases} 4.&-\pi \lt x \lt -\pi/2\\ -2.& -\pi/2 \lt x \lt \pi/2\\ 4.&\pi/2 \lt x \lt \pi \end {cases}$$

1. sketch the function,
2. from the sketch estimate the value of $a_0$,
3. analyse the fourier series sketch and determine whether or not $a_n$ or $b_n$ have values.

I can do 1 and 3 but not 2, I could calculate $a_0$ but not estimate it from the sketch.

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Answer 1

$a_0$ is the average value of the function over a period. How much of the period is the value $4$ and how much $-2$? You should be able to guess the average.

Answer 2

This is a sum of shifted rectangle functions - each of who's Fourier series coefficients is a phase shifted sinc. Simply compute the sinc terms for each of the rectangles and then add the series coefficients term by term. $a_0$ is just the mean on the interval: $$a_0 = \frac{4(\pi/2) -2(\pi) + 4(\pi/2)}{2\pi} = 1$$