  We give an algorithm to compute $N$ steps of a convolution quadrature
approximation to a continuous temporal convolution using only $O(N \log N)$
multiplications and $O(\log N)$ active memory. The method does not require
evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of
its Laplace transform, which is assumed sectorial.
  The algorithm can be used for the stable numerical solution with
quasi-optimal complexity of linear and nonlinear integral and
integro-differential equations of convolution type. In a numerical example we
apply it to solve a subdiffusion equation with transparent boundary conditions.
