Jürgen Prestin

Jürgen Prestin, Christian Wülker

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)} (r^2) r^{l} Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal basis of the space $L^{2}$ on $\mathbb{R}^{3}$ with Gaussian weight $\exp(-r^{2})$. 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 $\mathcal{O}(B^{4})$, $B$ being the respective bandlimit, while the number of sample points on $\mathbb{R}^{3}$ scales with $B^{3}$. This clearly improves the naive bound of $\mathcal{O}(B^{7})$. 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 $\mathcal{O}(B^{3} \log^{2} B)$ fast SGL Fourier transform and inverse, and an outlook on future developments.

Jürgen Prestin, Christian Wülker

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, constitute an orthonormal polynomial basis of the space $L^{2}$ on $\mathbb{R}^{3}$ with radial Gaussian weight $\exp(-r^{2})$. We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.

Illya M. Karabash, Jürgen Prestin

We address a parametric joint detection-estimation problem for discrete signals of the form $x(t) = \sum_{n=1}^{N} α_n e^{-i λ_n t } + ε_t$, $t \in \mathbb{N}$, with an additive noise represented by independent centered complex random variables $ε_t$. The distributions of $ε_t$ are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., the frequencies $λ_n$, their number $N$, and complex amplitudes $α_n$. For example, one of considered classes of noise is the following: $ε_t$ are independent identically distributed random variables with $\mathbb{E} (ε_t) = 0$ and $\mathbb{E} (|ε_t| \ln |ε_t|) < \infty$. The construction of estimators is based on detection of singularities of anti-derivatives for $Z$-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series. We discuss also decaying signals and the case of infinite number of frequencies.