Collocating Convolutions

An explicit method is derived for collocating either of the convolution integrals p(x) = fi f(x t)g(t)dt or q(x) = /*/(< x)g(t)dt, where x 6 (a, b), a subinterval of M . The collocation formulas take the form p = F(Am)% or q = F(Bm)g, where g is an w-vector of values of the function g evaluated at the "Sine points", Am and Bm are explicitly described square matrices of order m, and F(s) = ¡Qexp[-t/s]f(t)dt, for arbitrary c e [(b a), oo]. The components of the resulting vectors p (resp., q) approximate the values of p (resp., q) at the Sine points, and may then be used in a Sine interpolation formula to approximate p and q at arbitrary points on (a, b). The procedure offers a new method of approximating the solutions to (definite or indefinite) convolution-type integrals or integral equations as well as solutions of partial differential equations that are expressed in terms of convolution-type integrals or integral equations via the use of Green's functions. If u is the solution of a partial differential equation expressed as a f-dimensional convolution integral over a rectangular region B, and if u is analytic and of class Lipa on the interior of each line segment in B , then the complexity of computing an «-approximation of u by the method of this paper is c?([\o%(e)\2v+1).