Orthogonality condition for the 1st-kind Bessel function J_m(adsbygoogle = window.adsbygoogle || []).push({});

$$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$

where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st kind Bessel functions of order m:

$$f(r)=\sum_n a_n\cdot J_m(\alpha_{m,n}*r/R),$$ with $$a_n=\frac{2}{R^2J_{m \pm 1}^2(\alpha_{mn})}\int_0^R J_m(\alpha_{mn}*r/R) f(r) r dr$$. However what if the original function is something like:

$$f(r)=\sum_n a_n\cdot J_m(\alpha_{m,n}*r/R)+b_n\cdot J_{m+1}(\alpha_{m+1,n}*r/R),$$

which combines Bessel functions of different orders. The final question is can any function be decomposed into Bessel functions someshow, like FFT/IFFT(fast fourier transform / inverse fast Fourier transform)?

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# A Bessel decomposition for arbitrary function

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