How to mix 2 Fourier tables.

Question

I am currently working on a homework problem in relation to the Fourier series. Here are the function's properties:

f(x)= {0, 0<x<1} {1, 1<x<2} {0, 2<x<3}

We are asked to find the Fourier equation from the Fourier tables. However, as you can see, there are 3 conditions, and all of the given Fourier tables only have 2 conditions. Our teacher suggested that we mix 2 of the Fourier table, however, I have no idea where to start or what to do.

I tried to rework the function in order to have only 2 conditions as below:

f(x)= {1, 1<x<2} {0, else}

But no luck, once again the calculated function is not corresponding to the one given.

Thank you for your help.

Answer 1

Notice that you know the Fourier transform is linear, meaning that the transform of the sum (or difference) of two functions is the sum (or difference) of their two transforms. So, can we write this $f$ as the sum (or difference) of two functions whose transforms we know?

This $f$ has a step up followed by a step down. Can we write this function as a step up minus a later step up? (Think a bit before revealing the below by hovering your pointer over it.)

$$ f(x) = \left( \begin{cases} 0 ,& 0 < x \leq 1 \\ 1 ,& 1 < x \end{cases} \right) - \left( \begin{cases} 0 ,& x < 2 \\ 1 ,& 2 \leq x < 3 \end{cases} \right) $$

We should not be concerned about the difference between "${}<{}$" and "${}\leq{}$", since the integral in the definition of the transform is blind to it.