Can we have different methods to estimate elements from Lp spaces?

Question

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the most/only used method. Is it true that if we assume that the data set is an element of $L^2$ then after FFT we would obtain an element of $L^2$? Now if we form an auto-correlation sequence form the data set can we say that the auto-correlation sequence is an element of $L^1$. Therefore if we perform FFT on the auto-correlation sequence we do not get an element of $L^2$ but some other $L^p$ space (is it L infinity)? Any comment would be appreciated. Can we interpolate to other $L^p$ spaces from the above results?