Assume that we have a time series and we have calculate the corresponding auto-covariance function. Having the auto-covariance function we can calculate the corresponding power spectrum and having the power spectrum we can "restore" the auto-covariance function.
As far as I understood, to obtain the power spectrum we use Fourier transformation and to calculate the auto-covariance function from the power spectrum we use a back Fourier transformation. So, one can say that auto-covariance function and power spectrum are just time and frequency representation of the same function.
The first part of my question is about the interpretation of the power spectrum. For example, for the white-noise the power spectrum is a constant. Does it mean that the white noise can be understood as a sum of sinusoid functions with different frequencies and the same amplitude (the density distribution function of the frequency is uniform)?
I think that the answer is "No". I think that the auto-covariance function should be understood as an infinite sum of sinusoids.
The second part of the question is about use of the power-spectrum. Assume that we have it. Can we use it to predict time series? Or how do we use it?