Gaussian-convolution-invariant shell approximation to spherically-symmetric functions

TL;DR

This paper introduces a new class of functions, Omega_N, that approximate spherically-symmetric functions and are invariant under Gaussian convolution, enabling efficient modeling of physical scalar fields.

Contribution

The authors develop explicit formulas for Omega_N functions in 1, 2, and 3 dimensions, demonstrating their utility in representing oscillating spherical functions with convolution invariance.

Findings

Omega_N functions are analytically expressed for N=1, 2, 3.

Representation of oscillating functions using Omega_N is computationally efficient.

Application examples include modeling electron density and scattering potentials.

Abstract

We develop a class of functions Omega_N(x; mu, nu) in N-dimensional space concentrated around a spherical shell of the radius mu and such that, being convoluted with an isotropic Gaussian function, these functions do not change their expression but only a value of its 'width' parameter, nu. Isotropic Gaussian functions are a particular case of Omega_N(x; mu, nu) corresponding to mu = 0. Due to their features, these functions are an efficient tool to build approximations to smooth and continuous spherically-symmetric functions including oscillating ones. Atomic images in limited-resolution maps of the electron density, electrostatic scattering potential and other scalar fields studied in physics, chemistry, biology, and other natural sciences are examples of such functions. We give simple analytic expressions of Omega_N(x; mu, nu) for N = 1, 2, 3 and analyze properties of these functions. Representation of oscillating functions by a sum of Omega_N(x; mu, nu) allows calculating distorted maps for the same cost as the respective theoretical fields. We give practical examples of such representation for the interference functions of the uniform unit spheres for N = 1, 2, 3 that define the resolution of the respective images. Using the chain rule and analytic expressions of the Omega_N(x; mu, nu) derivatives makes simple refinement of parameters of the models which describe these fields.