  In the 1950s L. Schwartz proved his famous impossibility result: for every k
in N there does not exist a differential algebra (A,+,*,D) in which the
distributions can be embedded, where D is a linear operator that extends the
distributional derivative and satisfies the Leibnitz rule (namely
D(u*v)=Du*v+u*Dv) and * is an extension of the pointwise product on the
continuous functions. In this paper we prove that, by changing the requests, it
is possible to avoid the impossibility result of Schwartz. Namely we prove that
it is possible to construct an algebra of functions (A,+,*,D) such that (1) the
distributions can be embedded in A in such a way that the restriction of the
product to the C^{1} functions agrees with the pointwise product, namely for
every f,g in C^{1} we have
  {\Phi}(fg)={\Phi}(f)*{\Phi}(g), and (2) there exists a linear operator
D:A\rightarrow A that extends the distributional derivative and satisfies a
weak form of the Leibnitz rule. The algebra that we construct is an algebra of
restricted ultrafunction, which are generalized functions defined on a subset
{\Sigma} of a non-archimedean field K (with {\Sigma}\subseteq R\subseteq K) and
with values in K. To study the restricted ultrafunctions we will use some
techniques of nonstandard analysis.
