A New Algebra of Periodic Generalized Functions

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 $\mathcal D^{'(n)}_{2\pi}$ 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 $\mathcal D^{'(n)}_{2\pi}$.