  The Legendre-based ultraspherical spectral method for ordinary differential
equations is combined with a formula for the convolution of two Legendre series
to produce a new technique for solving linear Fredholm and Volterra
integro-differential equations with convolution-type kernels. When the kernel
and coefficient functions are sufficiently smooth then the method is
spectrally-accurate and the resulting almost-banded linear systems can be
solved with linear complexity.
