Variational derivation and extensions of distributed approximating functionals

Extensions of the ''Distributed Approximating Functional'' (DAF) approach to approximating functions and their derivatives are given. The method, although inherently approximate, can be made arbitrarily accurate, numerically stable, and computationally efficient by appropriate choice of parameters. It also provides approximate representations of quantum operators which are analytic and which can be made arbitrarily accurate. Differences between the DAFs and more standard basis set approaches are discussed in order to clarify the properties of the DAFs. Some illustrative examples are given.