Kernel Identities and Vectorial Regularization

TL;DR

This paper introduces 'vectorial regularization' as a novel method to prove kernel identities, successfully deriving both known and new identities in the theory of distributions and tensor products.

Contribution

The paper presents a new technique called vectorial regularization for proving kernel identities, expanding the set of known identities in distribution theory.

Findings

Established new kernel identities using the method.

Reproved known kernel identities with the new approach.

Demonstrated the effectiveness of vectorial regularization in distribution theory.

Abstract

We present the method of "vectorial regularization" to prove kernel identities. This method is applied to derive both known kernel identities, e.g. $\dot{\mathcal{B}}_{xy}=\dot{\mathcal{B}}_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}_y$, $\mathcal{D}'_{L^1,xy}=\mathcal{D}'_{L^1,x}\widehat{\otimes}_\pi\mathcal{D}'_{L^1,y}$, as well as new ones: $\dot{\mathcal{B}}'_{xy}=\dot{\mathcal{B}}'_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}'_y$ and $\mathcal{D}_{L^1,xy}=\mathcal{D}_{L^1,x}\widehat{\otimes}_\pi\mathcal{D}_{L^1,y}$.