On the correctness of finite-rank approximations by series of shifted Gaussians

TL;DR

This paper proves the correctness of finite-rank linear system approximations for Gaussian series interpolation, overcoming numerical difficulties of previous methods by deriving explicit formulas to ensure system invertibility.

Contribution

It introduces a new approach based on finite-rank approximations and proves its correctness through explicit determinant formulas, advancing Gaussian series interpolation methods.

Findings

Finite-rank linear system approximation is correct.

Explicit determinant formula ensures system invertibility.

Addresses numerical difficulties in Gaussian interpolation.

Abstract

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite-rank approximations by linear systems. The main result for this approach is to establish correctness of the finite-rank linear system under consideration. And the main result of the paper is to prove correctness of the finite-rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non-zero.