Schwartz核的精确反卷积：从多项式自同构到温分布中的递归反卷积
Exact Deconvolution for Schwartz Kernels: From Polynomial Automorphisms to Recursive Inversion in Tempered Distributions
AI总结：
本文提出一种基于迭代前向卷积的精确反卷积方法，通过证明偶核与多项式卷积的自同构性，推导出有限多项式空间的精确代数反演，并扩展到无限维函数空间，提供新的迭代数值公式用于Weierstrass变换的反演。
摘要：
本文构建了一种理论严谨的显式反卷积方法，完全依赖于迭代前向卷积，可进行数值实现。我们首先证明，与偶Schwartz核卷积在有限次多项式向量空间上作用为自同构。利用核的奇偶性，我们推导出该空间的精确代数反演，以有限线性组合的重复卷积唯一表示。本文的核心贡献是将这种代数反演扩展到无限维函数空间，包括L¹(R)、L²(R)、Schwartz空间S(R)以及温分布空间S'(R)。通过将有限和多项式反演公式取极限，我们证明任意函数或分布与Schwartz核卷积后可在相应拓扑中精确恢复。所得反演是显式可计算的递归卷积线性组合序列的极限。作为主要应用，此极限提供了一种新的迭代数值公式用于Weierstrass变换的反演。通过绕过传统数值不稳定反演技术，本文方法提供了一个数学精确且数值稳健的计算信号恢复算法。
英文摘要：
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.
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