Schwartz Functions and Compactifications

TL;DR

This paper extends Schwartz functions to the real projective space compactification, generalizing earlier results about smooth extensions at infinity in Euclidean spaces.

Contribution

It introduces a new characterization of Schwartz functions in the context of real projective space compactifications, expanding the understanding of smooth function extensions.

Findings

Schwartz functions can be extended smoothly to real projective spaces.

The paper establishes a correspondence between Schwartz functions and smooth functions on the compactification.

It generalizes classical results from Euclidean spaces to projective spaces.

Abstract

In the early 20th century, Laurent Schwartz observed that we can identify functions that extend smoothly to the point at infinity of one-point compactifications of Euclidean spaces. We show a similar result for a different compactification of Euclidean spaces, namely, the real projective spaces.