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?

enter image description here

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.

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    $\begingroup$ What you have written is essentially correct, except that the frequency components are at 1, 2, and 5 radians per second, not Hertz (cycles per second), assuming $x$ represents seconds. If you want you want 1, 2, and 5 Hertz, then your functions should be $\sin(2\pi x)$, $\sin(4\pi x)$, and $\sin(10\pi x)$. And if you want the x-axis in your frequency domain plot (figure 3) to be Hertz, then define your Fourier transform as follows: $\int_{-\infty}^{\infty}f(x)e^{-i2\pi \xi x} dx$. $\endgroup$ – Bungo Aug 28 '14 at 22:59
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    $\begingroup$ As I understand Fourier transforms, this makes perfect sense. I can add that if you remove the absolute value signs around $\hat f(\xi)$, the argument of that complex number tells you the phase of the $\xi$-component of the wave. $\endgroup$ – Arthur Aug 28 '14 at 23:01
  • $\begingroup$ +1 Thank you guys for your help! =) Appreciate your time and effort a lot! $\endgroup$ – jjepsuomi Aug 28 '14 at 23:03

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