I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct understanding of the Fourier transform. So my question/problem is that I would like to have a verification that the function and its frequency graph below make sense. Here goes my example:
Suppose I have the $2\pi$-periodic function
$$f(x) = 2\sin(x) + 3\sin(4x) + 4\sin(5x) $$
and its Fourier transform is defined as:
$$\mathscr{F}\left[ f(x) \right] = \hat{f}(\xi) = \int_{-\infty}^{\infty}f(x)e^{-i\xi x}\;dx, \;\;\;\;\;\;\;\; \xi = 0,1,2,...,$$
where $\xi$ is the frequency, so $\xi=1$, means the $1Hz$ frequency, etc. So $\hat{f}(\xi)\neq 0,$ when $\xi =1,4,5$ and $0$ otherwise, because I have only $1Hz, 4Hz$ and $5Hz$ components in my function. Because $\hat{f}(\xi )$ is a complex number I need to take the absolute value $\left| \hat{f}(\xi)\right|$ and this value corresponds to the amplitudes of the sinusoidal components. In the below picture I have drawn the function in time domain and frequency domain. Does it make any sense?
Hope my question is clear =) Thnx for any help! =) P.S. if the image is hard to see, just right-click --> view image / see in other window etc.