Lorentz-invariant pseudo-differential wave equations

We define in a consistent way nonlocal pseudo-differential operators acting on a space of analytic functionals. We discuss the relation;of our method to other definitions for nonlocal operators. We show how to solve homogeneous and inhomogeneous equations associated with nonlocal pseudo-differential operators, and we extend the formalism to d-dimensional space-time solving, in particular, the fractional wave and Klein-Gordon equations. We also study in this context the nonlocal equations obtained from effective QED when integrating over the fermionic degrees of freedom.