Suppose we have a sine voltage supplied to a resistive load. When I perform a Fourier transform to this signal in the frequency domain what do I get? All values of g(a) will be 0 except for the frequency of my sine signal?
Yes, that is correct. Since there are no harmonics in a sine (other than the fundamental) there will only be that frequency in the transform.
Figure 1. This fabulous illustration of the Fourier Transform by Lucas V. Barbosa on Wikipedia's Fourier transform page shows the transformation of a periodic waveform from the time domain to the frequency domain. The frequency plot shows the relative strength of the harmonics with clarity that could not be obtained from staring at the time plot.
- It should be apparent that the more square the time domain waveform is then the more harmonics you will have and these should be visible in the frequency domain.
- It should also be clear that the amplitude decreases with the increasing frequency.
For your question it should also be clear that you only have the fundanmental (largest blue sine signal) in the composite signal and therefore only have the first frequency line in the Fourier transform.
The sine function can be Fourier transformed only using distributions and the result is a summation of two Dirac's delta distributions centered at +f and -f, multiplied by i where i is the imaginary unit.
That distribution is imaginary.
Take a look here: