# Bessel decomposition for arbitrary function

• A
Orthogonality condition for the 1st-kind Bessel function J_m
$$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$
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)?

Orodruin
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A priori, you could expand any Bessel function ##J_n(\alpha_{nk'}r/R)## in terms of the Bessel functions ##J_m(\alpha_{mk}r/R)## for any fixed ##n## and ##m##. It is just a matter of using a different basis for your function space.

Thanks for reply. I did NOT know Bessel function of order n can be expanded into Bessel functions of order m. Could you please give me an identity?

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Orodruin
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While it can be done, it is not clear why you would want to do this. In general, Bessel function expansions will arise naturally as Bessel functions of different orders are eigenfunctions of different Sturm-Liouville operators. It is therefore usually quite clear what Bessel functions you should use for your expansion.

In my case, I do not know which order of Bessel functions should be used as the basis functions.

In many other cases, the original profile ##f(r)## contains Bessel functions of different orders. I am wondering how can I decompose it with ##J_m(r)## with fixed ##m =1##?

Orodruin
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Perhaps it would be a good idea to state the type of problem you are looking to solve. I think it would help us understand what you need better.

What I am doing now is simple. I have a random series ##f(r)##, with boundary condition ##f(0)=0## and ##f(1)=0##. I want to approximate it using ##J_1(\alpha_{1n}r)##, which satisfies the B.Cs.

I had a plan to solve a system of ##nr## linear equations numerically, where ##nr## is the number of grid points along ##r## direction, as follows:
$$\sum_{i=1}^{nr} a_i J_1(\alpha_{1,i}\cdot r_i)=f(r_i),$$ for ##i=1,2,3...,nr##
Not sure if it is what you are going to suggest?

Not sure how do it analytically using orthogonality and other identities of Bessel functions.