On the Shape Fields Finiteness Principle

TL;DR

This paper advances the understanding of shape fields by improving finiteness constants for certain smooth function classes and extending the finiteness principle to vector-valued cases, enhancing the theoretical framework in this area.

Contribution

It improves the finiteness constant for existing shape fields finiteness principles and extends these principles to vector-valued functions.

Findings

Enhanced finiteness constants for $C^m$ and $C^{m-1,1}$ classes.

Extended shape fields finiteness principle to vector-valued functions.

Strengthened theoretical foundation for shape fields and smooth function selection.

Abstract

In this paper, we improve the finiteness constant for the finiteness principles for $C^m(\mathbb{R}^n,\mathbb{R}^d)$ and $C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)$ selection proven by Fefferman, Israel, and the second author and extend the more general shape fields finiteness principle to the vector-valued case.