Explicit Formulas for the Inversion of the Convolution of Polynomials and Arbitrary Functions with Schwartz Kernels

Convolution serves as a powerful operation for the regularization of functions. While polynomials inherently possess smoothness, it is particularly interesting to investigate their behavior under convolution. This interest stems from the fact that numerous engineering and physical phenomena can be modeled through such operations, including weighted averages, blurring effects and convolutional integral equations. In this work, we show that under certain mild conditions, the convolution with any even Schwartz function acts as an automorphism on the vector space of finite-order polynomials. We derive explicit equations for the inverse operation of this convolution, which are numerically simple to implement. In addition, we extend the deconvolution with (not necessarily even) Schwartz functions to a broader class of functions, including $L^1(\mathbb{R})$, $L^2(\mathbb{R})$, the Schwartz space and tempered distributions. Specifically, we establish a explicit rigorous formula for the deconvolution of a function or distribution that has been convolved with a Schwartz function, being a particular example the Weierstrass Transform. For the latter, we show that any Schwartz function and tempered distribution that has been transformed, can be recovered, in their respective topologies, by the limit of a sequence of linear combination of recursive convolutions. This provides a new formula for the inverse of the Weierstrass Transform that can be numerically implemented.