On dense subspaces in a class of Fr\'echet function spaces on R^n

TL;DR

This paper presents a theorem for constructing dense subspaces in Fréchet function spaces on R^n, facilitating problem-specific function approximation by identifying subspaces of a particular form.

Contribution

It introduces a general theorem that characterizes dense subspaces in common Fréchet function spaces, unifying classical approximation results.

Findings

Provides a method to construct dense subspaces of the form {pf_0 | p∈P}

Unifies classical approximation theorems like Stone-Weierstrass and Hermite function completeness

Applicable to various function spaces on R^n

Abstract

When dealing with concrete problems in a function space on R^n, it is sometimes helpful to have a dense subspace consisting of functions of a particular type, adapted to the problem under consideration. We give a theorem that allows one to write down many of such subspaces in commonly occurring Fr\'echet function spaces. These subspaces are all of the form $\{pf_0 | p\in{\cal P}\}$ where $f_0$ is a fixed function and ${\cal P}$ is an algebra of functions. Classical results like the Stone-Weierstrass theorem for polynomials and the completeness of the Hermite functions are related by this theorem.