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

Abstract

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.

Figures (3)

• Figure 1: Representation of a numerical validation of Theorem \ref{['thm:ConvolutionIsAnIsomorphismInPolynomials']}. The left picture represents as a solid black line the original polynomial $p$ which is the $n=50$ order Taylor polynomial of the function $x \in \mathbb R \mapsto \sin(5x) + \sin(3x)$. It also represents as a blue solid line its convolution $p * \varphi_{\varepsilon}$ with convolution kernel $s\mapsto \varphi(x) = \mathop{\mathrm{ind}}\nolimits_{(-1,1)}\exp(-1/(1-x^2))$ and parameter $\varepsilon = 0.9$. The dashed red line represents $T_{\varepsilon}^{-1}(p * \varphi_{\varepsilon})$ computed using equation \ref{['eq:InverseOfConvolution']}. The right picture represents the error between the original polynomial and the inverse of the convolution.
• Figure 2: Representation of the numerical validation of Theorem \ref{['thm:GeneralizationToSchwartzFunctionsAndDistributions']} under ideal conditions. Left picture represents in black the original function $t \mapsto f(t) = \sin(5t)+\sin(3t)$, in blue its convolution $T_{\varepsilon}(f)$ with the Gaussian Kernel $\varphi$ with $\varepsilon = 0.55$, and as dashed red line its reconstruction $T_{\varepsilon,n}^{-1}(T_{\varepsilon}(f))$ using Equation \ref{['eq:InverseOfConvolutionArbitrary']} for $n= 90$. Right picture represents in black the Fast Fourier Transform (FFT) of $f$, in green the FFT of $\varphi$, in blue the FFT of $T_{\varepsilon}(f)$ and in red the FFT of the reconstructed function $T_{\varepsilon,n}^{-1}(T_{\varepsilon}(f))$.
• Figure 3: Representation of the numerical validation of Theorem \ref{['thm:GeneralizationToSchwartzFunctionsAndDistributions']} under non ideal conditions. Top and bottom left pictures represent in black the original function $t \mapsto f(t) = \sin(5t)+\sin(3t)$, with a blue line $T_{\varepsilon}(f)$ with $\varepsilon = 0.55$ and $\varphi$ the Gaussian Kernel, as a yellow line $T_{\varepsilon}(f) + \eta$, where $\eta$ is noise generated using a normal distribution with zero mean and variance 0, and with a red line $T_{\varepsilon,n}^{-1}(T_{\varepsilon}(f)+\eta)$ where $n =90$. Top right picture depicts as a black line the FFT of $f$, as a yellow line the FFT of $T_{\varepsilon}(f)+\eta$ and as a red line $T_{\varepsilon,n}^{-1}(T_{\varepsilon}(f)+\eta)$, i.e., the application of Equation \ref{['eq:InverseOfConvolutionArbitrary']} to $T_{\varepsilon}(f)+\eta$. The bottom right picture shows with a black line the FFT of $f$, with a green line the FFT of the used filter $t \mapsto h(t) = 2\mathop{\mathrm{sinc}}\nolimits(2t)$, with a yellow line the FFT of $T_{\varepsilon}(f)+\eta$ and with a red line the FFT of $T_{\varepsilon,n}^{-1}(T_{\varepsilon}(f)+\eta) * h$, that is, $g := T_{\varepsilon}(f) + \eta$.

Theorems & Definitions (19)

• Definition 1
• Remark 1
• Theorem 1
• proof
• Corollary 1
• proof
• Remark 2
• Corollary 2
• proof
• Lemma 1