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.

翻译：我们发展了一类N维空间中的函数Ω_N(x; μ, ν)，这些函数集中在半径为μ的球壳周围，并且在与各向同性高斯函数进行卷积时，其表达式保持不变，仅改变其"宽度"参数ν的值。各向同性高斯函数是Ω_N(x; μ, ν)在μ = 0时的特例。由于其特性，这些函数成为构建平滑连续球对称函数（包括振荡函数）近似的高效工具。在物理、化学、生物学等自然科学中研究的电子密度、静电散射势及其他标量场的有限分辨率图谱中的原子图像，就是此类函数的实例。我们给出了N = 1, 2, 3时Ω_N(x; μ, ν)的简单解析表达式，并分析了这些函数的性质。通过Ω_N(x; μ, ν)的线性组合表示振荡函数，能够以与相应理论场相同的计算成本生成畸变图谱。我们以N = 1, 2, 3时均匀单位球体的干涉函数为例，展示了这种表示方法的实际应用，这些函数决定了相应图像的分辨率。利用链式法则和Ω_N(x; μ, ν)导数的解析表达式，可以简便地优化描述这些场的模型参数。