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Is there any theorem that can be used (with appropriate assumptions) to prove that the Fourier coefficients of a given sequence (binary valued in my case) have to be at least something.

For example, a potential claim can be: There is a constant c that for any binary function of length n (e.g. $1010111001101$ for $n=13$) $|F_n|^2>=c/n^3$. In other words for a fixed $n$ we cannot cook up a binary function that has arbitrarily small nth Fourier coefficient.

An answer is given in the following question: Bounds for Fourier series Seems very relevant. Especially the part that says

Conversely, if |f^s|≤C|s|−k for some k≥2, then f is continuous with k−2 continuous derivatives. But there is nothing you can say about the size of f. For instance, if f=M is constant, then f^s=0 for all s≠0.

My problem is that my function is not continuous. More importantly, what is this theorem called? How can I cite it?

Eri
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1 Answers1

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Since you have a discrete time input sequence and are asking about discrete frequency domain coefficients, I can only assume that the transform you are interested in is the Discrete Fourier Transform (DFT).

In the case of the DFT, the example claim is certainly false, unless $c=0$. a A binary sequence of all 1's demonstrates that your example claim is false. This sequence will have a DFT that is 0 in every frequency bin, except the bin corresponding to 0 Hz.

In Octave or Matlab simply type:

fft((ones(1,13))

to see for yourself.

Since the DFT is a linear operator, you will get the same sort of result, if the values in the discrete time sequence are not 1; just as long as all the values in the input discrete time sequence are the same.

Parseval's Theorem (or Plancherel's Theorem) provides a specific relationship between the discrete time values and discrete frequency values, but not for any one individual discrete frequency value. https://en.wikipedia.org/wiki/Parseval%27s_theorem

Andy Walls
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