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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 almost periodicity of Bessel functions, generalizing the above formula.

Would it be possible to expand such an almost periodic function in a generalized Fourier series?

More in detail, is it legitimate to write the following relation?

$$ J_{0}(kz)\sim\sum_{m=-\infty}^{+\infty}\varphi_m e^{i\lambda_mz} $$

where the approximated expansion holds in an interval centered around $z=0$ and extends for a few quasi-periods.

If so, how is $\lambda_m$ calculated?

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    $\begingroup$ Almost periodic function cannot tend to zero as $x\to\infty$. The series you wrote never tends to zero as $z\to\infty$ on the real line. And Bessel functions do. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 8, 2020 at 13:16
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    $\begingroup$ I think this question could make sense in the following way. Is there an almost periodic function that asymptotically approximates $x^{1/2} J_0(x)$ with better error than $O(1/x)$, which is already accomplished by $(\sin x + \cos x)/\sqrt{\pi}$? If not, could this error be improved on some interval that doesn't extend all the way to infinity, but still gets asymptotically large? $\endgroup$
    – Igor Khavkine
    Commented Apr 8, 2020 at 13:29
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    $\begingroup$ Alessandro, do you need this interval to be "large" in any way? If not, fix your favorite interval, then compute the usual Fourier series on that interval. $\endgroup$
    – Igor Khavkine
    Commented Apr 8, 2020 at 14:11
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    $\begingroup$ @Alessandro Zunino: On any finite interval, you can approximate any reasonable function by a partial sum of its Fourier series. Or a series of exponentials provided that they are complete on this interval. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 9, 2020 at 0:43
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    $\begingroup$ Functions vanishing at infinity fall into the class of null weakly almsot periodic functions. These are exactly the weakly almost periodic functions with vanishing Fourier--Bohr coefficients. In particular, in the sense of Besicovitch or Weyl almsot periodicity, they are the same as the 0 function. So Besicovitch or Weyl almost periodicity will definitely NOT give you the coefficients. @YemonChoi $\endgroup$
    – Nick S
    Commented Jun 24, 2020 at 17:43

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Maple gave me this... $$ J_0(x) = \left( {\frac {\sin \left( x \right) }{\sqrt {\pi}}}+{\frac {\cos \left( x \right) }{\sqrt {\pi}}} \right) x^{-1/2}+ \left( -{\frac {\cos \left( x \right) }{8\sqrt {\pi}}}+{\frac {\sin \left( x \right) }{8\sqrt {\pi}}} \right) x^{-3/2} \\+ \left( -{\frac {9\,\sin \left( x \right) }{128\,\sqrt {\pi}}}-{ \frac {9\,\cos \left( x \right) }{128\,\sqrt {\pi}}} \right) x^{-5/2}+ \left( {\frac {75\,\cos \left( x \right) }{ 1024\,\sqrt {\pi}}}-{\frac {75\,\sin \left( x \right) }{1024\,\sqrt { \pi}}} \right) x^{-7/2} \\+ \left( {\frac {3675\, \sin \left( x \right) }{32768\,\sqrt {\pi}}}+{\frac {3675\,\cos \left( x \right) }{32768\,\sqrt {\pi}}} \right) x^{-9/2} \\+ \left( -{\frac {59535\,\cos \left( x \right) }{262144 \,\sqrt {\pi}}}+{\frac {59535\,\sin \left( x \right) }{262144\,\sqrt { \pi}}} \right) x^{-11/2}+o(x^{-11/2}) $$ as $x \to \infty$

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  • $\begingroup$ Thank you very much. Your Maple result looks nice, but I am looking for something like a Generalized Fourier series, as those of Besicovitch almost periodic functions (en.wikipedia.org/wiki/…). The multiplication term $x^{-1/2-n}$ looks like a big departure from an expansion in "discrete frequencies". $\endgroup$
    – DrManhattan
    Commented Apr 8, 2020 at 11:48
  • $\begingroup$ @AlessandroZunino Besicovitch AP seems to be a red herring if you merely want an approximation in some interval around $x=0$. As pointed out above, AP functions in the sense of Bohr cannot tend to zero at infinity, and I think the same also applies to the Besicovitch $B_p$-spaces $\endgroup$
    – Yemon Choi
    Commented Apr 8, 2020 at 21:04

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