Return to Question

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?

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

If so, how is $\lambda_m$ calculated?

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?

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?

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

If so, how is $\lambda_m$ calculated?

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?

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

If so, how is $\lambda_m$ calculated?

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?

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

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?

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

If so, how is $\lambda_m$ calculated?