A New Algebra of Periodic Generalized Functions

Abstract

Let $n$ denote a strictly positive integer. We construct a complex differential algebra ${\mathcal{G}}_n$ of so-called $2\pi$-periodic generalized functions. We show that the space ${\mathcal{D}}_{2\pi }^{{}^{\prime }(n)}$ of $2\pi$-periodic distributions on ${\mathbb{R}}^n$ can be canonically embedded into ${\mathcal{G}}_n$. Next we lay the foundation for calculation in ${\mathcal{G}}_n$. This algebra ${\mathcal{G}}_n$ enables one to solve, in a canonical way, differential problems with strong singular periodic data which have no solution in ${\mathcal{D}}_{2\pi }^{{}^{\prime }(n)}$.