Subspaces with Equal Closure

Abstract

We take a new and unifying approach toward polynomial and trigonometric approximation in topological vector spaces used in analysis on R n. The idea is to show in considerable generality that in such a space a module, which is generated over the polynomials or trigonometric functions by some set, necessarily has the same closure as the module which is generated by this same set, but now over the compactly supported smooth functions. The particular properties of the ambient space or generating set are, to a large degree, irrelevant for these subspaces to have equal closure. This translation—which goes in fact beyond modules—allows us, by what is now essentially a straightforward check of a few properties, to replace many classical results in various spaces by more general statements of a hitherto unknown type. Even in the case of modules with one generator the resulting theorems on, e.g., completeness of polynomials are then significantly stronger than the classical statements. This extra precision stems from the use of quasi-analytic methods (in several variables) rather than holomorphic methods, combined with the classification of quasi-analytic weights. In one dimension this classification, which then involves the logarithmic integral, states that two well-known families of weights are essentially equal. As a side result we also obtain an integral criterion for the determinacy of multidimensional measures which is less stringent than the classical version. The approach can be formulated for Lie groups and this interpretation then shows that many classical approximation theorems are “actually” theorems on the unitary dual of R n, thus inviting to a change of paradigm. In this interpretation polynomials correspond to the universal enveloping algebra of R n and trigonometric functions correspond to the group algebra. It should be emphasized that the point of view, combined with the use of quasi-analytic methods, yields a rather general and precise ready-to-use tool, which can very easily be applied in new situations of interest which are not covered by this paper.