  This paper is second in a series of three papers; the first of which is
"Summation Formulas, from Poisson and Voronoi to the Present"
(math.NT/0304187), and the third of which is "Automorphic Distributions,
L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an
expository paper, while the third proves a Voronoi-style summation formula for
the coefficients of a cusp form on GL(3,Z)\GL(3,R). This present paper contains
the distributional machinery used in the third paper for rigorously deriving
the summation formula, and also for the proof of the GL(3)xGL(1) converse
theorem given in the third paper. The primary concept studied is a notion of
the order of vanishing of a distribution along a closed submanifold.
Applications are given to the analytic continuation of Riemann's zeta function;
degree 1 and degree 2 L-functions; the converse theorem for GL(2); and a
characterization of the classical Mellin transform/inversion relations on
functions with specified singularities.
