Towards a~better understanding of 𝒞𝑝 functions and definable 𝒞𝑝 functions of several variables
Year: 2025
Type: article
Abstract: Building on some classical techniques in approximation theory of functions of one variable, we obtain new results concerning C p \mathcal{C}^{p} functions and definable (e.g., semialgebraic) C p \mathcal{C}^{p} functions of several variables. The starting point for our work is the following vague problem: can C p \mathcal{C}^{p} smoothness be tested on lines in R N \mathbb{R}^{N} parallel to the coordinate axes? Specific aspects of this problem are investigated both in general setting and in o-minimal setting. Of course, the behaviour of a function is not examined separately on each line, but it should be uniformly controlled (for otherwise, we cannot expect any positive answers to the problem). All the main results of the paper are accompanied by numerous examples and counterexamples.
Author: Rafał Pierzchała
Institution: Jagiellonian University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W4407704490