A general form of 2D Fourier transform eigenfunctions
In this paper, the general form of the two-dimensional Fourier transform (2D FT) eigenfunctions is discussed. It is obtained from the linear combination of the 2D separable Hermite Gaussian functions (SHGFs). For example, the rotated Hermite Gaussian functions (RHGFs) for the rotated coordinate and the Laguerre Gaussian functions (LGFs) for the polar coordinate are two special cases of the general form. With the aid of the general form, we can achieve these 2D functions with perfect orthogonality. Finding the combination coefficients is equivalent to the multinomial expansion problem. Therefore, we can apply the fast Fourier transform and some recurrence relations to the coefficients. The computation cost is much less than the close-form coefficients, which is associated with the Jacobi polynomials.