I have a set of $53$ Fourier coefficients. The dc term is $0$. The $26$ positive frequency amplitudes (coefficients) are given below. The $26$ negative frequency amplitudes are the same.
$\{0.014451, 0.0095385, 0.0111566, 0.0323018, 0.0234022, 0.0104497, 0.0097798, 0.0168872, 0.0797642, 0.012786, 0.0094381, 0.012149, 0.0533323, 0.0185514, 0.0099585, 0.0101803, 0.02066, 0.040164, 0.0116112, 0.0094714, 0.0135438, 0.1592482, 0.0155471, 0.0096407, 0.0107726, 0.0270934\}$.
The values should be accurate up to the fourth or fifth decimal place.
Can anyone give me a closed form expression for these amplitudes? In general there is no reason to expect a closed form for the Fourier coefficients, (especially those that are obtained using Fourier transforms and arbitrary sample rates), but I was able to get a closed form for the phase angles which leads me to believe that there might be a closed form for the amplitudes as well.
The phase angles are given by Angle[$6n+k$] = $\dfrac{\pi(53-18k-2n)}{53}$, with $0 < 6n+k < 54$. Is it possible to have a closed form for the phase angles and not for the Fourier coefficients (amplitudes)?
Looking at the data graphically it is obvious that the amplitudes don't grow like $x$ or $1/x$; the lowest value occurs at #11. The amplitudes look more like a $\cot$ or $\cot^2$ function or maybe even a $\sec$ or $\csc$ function. Any suggestions on how to proceed would be greatly appreciated. Thanks.
