  We construct differential algebras in which spaces of (one-dimensional)
periodic ultradistributions are embedded. By proving a Schwartz impossibility
type result, we show that our embeddings are optimal in the sense of being
consistent with the pointwise multiplication of ordinary functions. In
particular, we embed the space of hyperfunctions on the unit circle into a
differential algebra in such a way that the multiplication of real analytic
functions on the unit circle coincides with their pointwise multiplication.
Furthermore, we introduce a notion of regularity in our newly defined algebras
and show that an embedded ultradistribution is regular if and only if it is an
ultradifferentiable function.

Functional Analysis