Schwartz functions on quasi-Nash varieties

TL;DR

This paper introduces the Quasi-Nash category, unifying Nash manifolds and algebraic varieties, and extends Schwartz functions, tempered functions, and distributions to this setting, preserving key properties.

Contribution

It defines the Quasi-Nash category and generalizes Schwartz and tempered functions and distributions within it, unifying previous frameworks.

Findings

Properties from affine spaces, Nash manifolds, and algebraic varieties hold in Quasi-Nash.

The category unifies different geometric objects under a common framework.

Extension of Schwartz and tempered functions to Quasi-Nash preserves key analytical properties.

Abstract

We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that properties that hold on affine spaces, Nash manifolds and algebraic varieties, also hold in this category.