Given an $n$ input array, is there an algorithm that is faster than Fast Fourier Transform if we are only interested in obtaining the maximum amplitude frequency? Looking at the Cooley–Tukey algorithm it seems there would be a way to "prune" frequencies that are guaranteed to not be of maximum amplitude. The application seems common enough that some other people might have already worked on a similar problem. I am both interested in algorithms that are time-complexity-wise faster and practically faster through small "shortcuts" arising from this particular problem. Is there such an algorithm?

How do I take the average of multiple fast Fourier transforms (ffts)? I have multiple audios that I want to take the fft of and then average these results to smooth out the random noise that appears in each audio. The ffts of the audios are shown in the graph below. However the problem I'm having is that I can't work out how to ensure that all the arrays I'm averaging have the same length and the same frequency spacing, because sampling the audios at a rate inversely proportional to their length makes them the same size, however it changes the frequency spacing of each audio sample. What I'm trying to do is find a way to average all these results so I can look for trends in the signal. The audios are from a particular species of birds, so averaging across multiple recordings would give me an insight into the frequencies at which this species communicates at and could be used to do the same for other species.

For homework I've to give an algorithm that computes the coefficients of the polynomial $p(x)^n$ in time $O(n\log n)$, where $p(x)$ is a polynomial of degree 7. As an hint I'm told to consider first the case where $n$ is a power of 2. My guess is that I should use FFT and somehow manipulate it with modulo operations (maybe using the fact that for even powers the unity roots of power $n$ are the same as for $n/2$, only twice) but I'm pretty clueless about this one.