Subspaces with Equal Closure

We develop a method to approach the problem of describing the closure in topological vector spaces in analysis on Rn of a module over the polynomials or over the span of the exponentials with imaginary spectral parameter. The method is not restricted to a particular type of space or to cyclic modules. It is easy to apply and extends naturally to “mixed” versions. The results generalize and simplify several classical theorems and at the same time show that the conditions in some classical theorems are too stringent. The fundamental observation is that it can under rather general conditions be shown that the closure of e.g. a module which is generated over the polynomials by a certain set, is equal to the closure of the module generated over the compactly supported smooth functions by this same set. The now translated problem of describing the closure of the second subspace is in general much better accessible than the original formulation. We establish the method and give multidimensional non-cyclic applications in the Schwartz space, spaces of test functions and distributions, general spaces of Bernstein type and various spaces (including Lp-spaces) associated with a locally compact subset of Rn. Cases which are already rather special still include e.g. dense translation invariant subspaces of the Schwartz space on Rn, the classical sufficient criterion for the Bernstein problem on the real line, density of polynomials in Lp-spaces on subsets of Rn and as a consequence the determinacy of a class of multidimensional probability measures as it can be concluded directly from the measure. The method shows that classical conditions on integrability or decay can often be relaxed, as e.g. with the density of polynomials in Lp-spaces and determinate measures. The reason for this is the use of quasi-analytic methods combined with our classification of multidimensional so-called quasi-analytic weights. One of the pertinent equivalences on the real line is that a weight (i.e. an arbitrary bounded non-negative function) w on R is quasi-analytic, i.e.