We know that a periodic function (e.g. a trigonometric function) has the property
$$ f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z $$
A Bessel function is not exactly periodic, because the value of the function roughly decreases after each oscillation. However, one could say that is not very far from being periodic. I would like to know if it is possible to express this quasi-periodicity of Bessel functions, generalizing the above formula.
Would it be possible to expand such an almost periodic function in a quasi-Fourier series?
More in detail, is it legitimate to write the following equation?
$$ J_{\nu}(kz)=\sum_{m=-\infty}^{+\infty}\varphi_m(\nu)e^{imkz} $$
The period estimate can be done by using autocorrelation sequence, where $R(\gamma)=\frac{\int_{n=a}^{n=b}f(x)f(x-\gamma)dx}{R(0)}$. Of course R(0) is just correlation of signal with itself without any delay, $\gamma$. After normalizing the sequence with R(0), the value of R(0) is taken as 1. So, now if you obtain the location of the second maximum in the sequence, it indicates the period because after that much delay we get a strong similarity. Then, you can try and approximate it with a Fourier series.