  Developments of nonlocal operators for modeling processes that traditionally
have been described by local differential operators have been increasingly
active during the last few years. One example is peridynamics for brittle
materials and another is nonstandard diffusion including the use of fractional
derivatives. A major obstacle for application of these methods is the high
computational cost from the numerical implementation of the nonlocal operators.
It is natural to consider fast methods of fast multipole or hierarchical matrix
type to overcome this challenge. Unfortunately the relevant kernels do not
satisfy the standard necessary conditions. In this work a new class of fast
algorithms is developed and analyzed, which is some cases reduces the
computational complexity of applying nonlocal operators to essentially the same
order of magnitude as the complexity of standard local numerical methods.
