Papers › Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions
Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions
Jürgen Prestin, Christian Wülker
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Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type Lₙ₋ₗ₋₁^((l + 1/2)) (r²) rˡ Yₗₘ(ϑ,φ), |m| ≤l < n ∈ℕ, Lₙ₋ₗ₋₁^((l + 1/2)) being a generalized Laguerre polynomial, Yₗₘ a spherical harmonic, constitute an orthonormal basis of the space L² on ℝ³ with Gaussian weight exp(-r²). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of 𝒪(B⁴), B being the respective bandlimit, while the number of sample points on ℝ³ scales with B³. This clearly improves the naive bound of 𝒪(B⁷). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true 𝒪(B³ log² B) fast SGL Fourier transform and inverse, and an outlook on future developments.
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