  We consider a class of linear integral operators with impulse responses
varying regularly in time or space. These operators appear in a large number of
applications ranging from signal/image processing to biology. Evaluating their
action on functions is a computation-ally intensive problem necessary for many
practical problems. We analyze a technique called convolution-product
expansion: the operator is locally approximated by a convolution, allowing to
design fast numerical algorithms based on the fast Fourier transform. We design
various types of expansions, provide their explicit rates of approximation and
their complexity depending on the time varying impulse response smoothness.
This analysis suggests novel wavelet based implementations of the method with
numerous assets such as optimal approximation rates, low complexity and storage
requirements as well as adaptivity to the kernels regularity. The proposed
methods are an alternative to more standard procedures such as panel
clustering, cross approximations, wavelet expansions or hierarchical matrices.
