COLLOCATING CONVOLUTIONS

An explicit method is derived for collocating either of the convolutio n integrals p(x) = integral(a)(x) f(x - t)g(t)dt or q(x) = integral(x) (b) f(t - x)g(t)dt, where x is an element of (a, b), a subinterval of R. The collocation formulas take the form p = F(A(m))g or q = F(B-m)g, where g is an m-vector of values of the function g evaluated at the ' 'Sinc points'', A(m) and B-m are explicitly described square matrices of order m, and F(s) = integral(0)(c) exp[-t/s]f(t)dt, for arbitrary c is an element of [(b - a), infinity]. The components of the resulting vectors p (resp., q) approximate the values of p (resp., q) at the Si nc points, and may then be used in a Sinc interpolation formula to app roximate 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 conv olution-type integrals or integral equations via the use of Green's fu nctions. If u is the solution of a partial differential equation expre ssed as a v-dimensional convolution integral over a rectangular region B, and if u is analytic and of class Lip(alpha) on the interior of ea ch line segment in B, then the complexity of computing an epsilon-appr oximation of u by the method of this paper is O([log(epsilon)](2v+2)).