ON THE COMPLETENESS OF (F(N THETA)) AND A RECIPROCAL THEOREM FOR ABSOLUTELY CONVERGENT DIRICHLET SERIES

Abstract

One of the basic questions in approximation theory is: given a Banach space B, and S a subset of B, under what conditions will S be "rich" enough to approximate any element of B by a finite linear combination of elements of S. The most important result of this type pertains to the case where B is the Banach space of continuous functions on a finite interval with the supremum norm, and S is the set of monomials {l,x,x 2, ••• }. In this case the famous Weierstrass Theorem asserts that any element of B can be uniformly approximated by a finite linear combination of elements of S, that is any continuous function on a finite interval can be uniformly approximated by a polynomial. (from Introduction)