What is the formula for Fourier series on the unit disk?

Question

Wikipedia gives a formula for two dimensional Fourier series on a square of length 2pi.

If I want to approximate functions on the unit disk do I just change the integration area from the square to the disk ?

Answer 1

Your formula would not work just by changing the domain of integration. In the same vein there is another useful basin given by the Fourier Bessel series. The idea is to go in polar coordinates and diagonalise the Laplacian by separation of variables. You write: $$ f(r,\phi) = \sum_{m,n\in\mathbb N^2} c_{mn}e^{im\phi}J_m(u_{mn}r) $$ with $u_{mn}$ the $n$th root of $J_m$. You can calculate the coefficients with: $$ c_{mn} = \frac{1}{\pi J_{m+1}(u_{mn})}\int_0^{+\infty}rdr\int_0^{2\pi}d\phi e^{-im\phi}J_m(u_{mn}r)f(r,\phi) $$

You can check that the basis functions $e_{mn}=e^{im\phi}J_m(u_{mn}r)$ are orthogonal and are eigenvectors of the Laplacian: $$\Delta=\partial_r^2+\frac{1}{r}\partial_r+\frac{1}{r^2}\partial_\phi^2$$

Hope this helps.