  The integration operators (*) $({\mathcal J}^+\,g)(x) = \int_a^x g(t) \, dt$
and (**) $({\mathcal J}^-\,g)(x) = \int_x^b g(t) \, dt$ defined on an interval
$(a,b) \subseteq {\mathbf R}$ yield new identities for indefinite convolutions,
control theory, Laplace and Fourier transform inversion, solution of
differential equations, and solution of the classical Wiener--Hopf integral
equations. These identities are are expressed in terms of ${\mathcal J}^\pm$\,
and they are thus esoteric. However the integrals (*) and (**) can be
approximated in many ways, yielding novel and very accurate methods of
approximating all of the above listed relations. Several examples are
presented, mainly using Legendre polynomial as approximations, and references
are given for approximation of some of the operations using Sinc methods. These
examples illustrate for a class of sampled statistical models, the possibility
of reconstructing models much more efficiently than by the usual slow
Monte--Carlo $({\mathcal O}(N^{-1/2})$ rate. Our examples illustrate that we
need only sample at $5$ points to get a representation of a model that is
uniformly accurate to nearly $3$ significant figure accuracy.
