Analytic Dirac approximation for real linear algebraic groups

TL;DR

This paper introduces an explicit Dirac sequence in the algebra of analytic vectors for real linear algebraic groups, providing a straightforward proof of the density of analytic vectors in certain representations.

Contribution

It presents an explicit Dirac sequence in A(G) and uses it to prove the density of analytic vectors in moderate growth representations.

Findings

Explicit Dirac sequence in A(G) constructed

Elementary proof of density of analytic vectors

Applicable to representations of moderate growth

Abstract

For a real linear algebraic group G let A(G) be the algebra of analytic vectors for the left regular representation of G on the space of superexponentially decreasing functions. We present an explicit Dirac sequence in A(G). Since A(G) acts on E for every Frechet-representation (\pi,E) of moderate growth, this yields an elementary proof of a result of Nelson that the space of analytic vectors is dense in E.