  We develop a nonlinear theory for infrahyperfunctions (also referred to as
quasianalytic (ultra)distributions by L. H\"{o}rmander). In the hyperfunction
case our work can be summarized as follows. We construct a differential algebra
that contains the space of hyperfunctions as a linear differential subspace and
in which the multiplication of real analytic functions coincides with their
ordinary product. Moreover, by proving an analogue of Schwartz's impossibility
result for hyperfunctions, we show that this embedding is optimal. Our results
fully solve an earlier question raised by M. Oberguggenberger.
