Toward a Unified Theory of Generalized Functions: Convergence

Generalized functions are usually treated as bounded linear functionals on spaces of test functions. Here they are considered as objects which ?convolute”? with test functions and satisfy a certain associativity condition. In this setting simple definitions are given for convergence of sequences of test functions and generalized functions, definitions which do not depend on particular properties of the space of test functions. It is shown that for the most commonly used spaces, these general definitions reduce to the standard ones.