Exact Deconvolution for Schwartz Kernels: From Polynomial Automorphisms to Recursive Inversion in Tempered Distributions

TL;DR

This paper develops an explicit, iterative deconvolution method for Schwartz kernels that can exactly invert convolutions in various function spaces, including distributions, offering a robust alternative to traditional ill-posed techniques.

Contribution

It extends algebraic inversion formulas from polynomials to infinite-dimensional spaces, enabling exact deconvolution of Schwartz kernels in distributions.

Findings

Derived an exact algebraic inverse for polynomial convolutions using kernel parity.

Extended the inversion to Schwartz space and tempered distributions.

Provided a new iterative numerical formula for the inverse of the Weierstrass Transform.

Abstract

In this work, we construct an explicit, theoretically rigorous deconvolution method that relies entirely on iterative forward convolutions, thus can be numerically implemented. We first prove that convolution with an even Schwartz kernel acts as an automorphism on the vector space of finite-degree polynomials. Exploiting the parity of the kernel, we derive an exact algebraic inverse for this space, expressed uniquely as a finite linear combination of repeated convolutions. The core contribution of this work extends this algebraic inversion to infinite-dimensional function spaces, including $L^1(\mathbb{R})$, $L^2(\mathbb{R})$, the Schwartz space $\mathscr{S}(\mathbb{R})$, and the space of tempered distributions $\mathscr{S}'(\mathbb{R})$. By passing the finite-sum polynomial inversion formula to the limit, we demonstrate that an arbitrary function or distribution convolved with a Schwartz kernel can be exactly recovered in its respective topology. The resulting inverse is an explicitly computable limit of a sequence of linear combinations of recursive convolutions. As a primary application, this limit provides a fundamentally new, iterative numerical formula for the inverse of the Weierstrass Transform. By bypassing traditional numerically ill-posed inversion techniques, our method offers a mathematically exact and numerically robust algorithm for computational signal recovery.