Spectral representation of functional and difference operators

The difference and differential-difference equations are special cases of the class of functional-differential equations (FDE). In particular, delay differential equations, widely used for modelling physical and other phenomena having non-local nature, belong to this class. The use of spectral methods for solution of such problems is difficult, since it requires approximation of difference and functional operators of various nature and of many different types. Here we propose a unified method of approximation of these operators for arbitrary sets of basis functions. The results have applications to the analytic number theory.