Extended automorphic forms

Many computations in the harmonic analysis of automorphic forms, especially concerning Eisenstein series, or, even worse, trace formulas, expose one to the danger that a naive formal approach leads to incorrect manipulation of expressions whose convergence is fragile or even volatile. At the same time it is often clear that some improved formalism can be correct, and may be more intelligible than a classical treatment. Indeed, the extreme technicality of strictly classical versions of many of these computations gives considerable impetus to consideration of alternatives.