Bessel functions in an Hermite-Gauss basis

¿Could somebody tell me how can i write a zero order bessel function in an Hermite-Gauss basis? Thanks

-

The Hermite-Gauss functions (suitably normalized) are orthonormal for an inner product defined by integration, just like $\{\sin(nx),\cos(mx)\}$ as $n,m$ run over positive integers. The coefficients of an expansion of a $J$-Bessel function in terms of Hermite-Gauss functions are computed by integrating the Bessel functions against the various Hermite-Gauss functions, in a way analogous to how a Fourier expansion is computed. You'll want to be looking at a table of integrals, or maybe Mathematica. (I'm not guaranteeing anything about the convergence of the series; I haven't thought about it that much.)