On optimal solutions of the Borel problem in the Roumieu case

TL;DR

This paper investigates the optimal solutions to the Borel problem within ultradifferentiable classes, comparing different solutions through a unified framework and establishing conditions for their equivalence in the Roumieu case.

Contribution

It provides a unified analysis of optimal solutions to the Borel problem for various ultradifferentiable classes using one-parameter families of weight sequences.

Findings

Relations among optimal solutions are established.

Conditions for equivalence of solutions are identified.

Unified framework facilitates comparison across classes.

Abstract

The Borel problem for Denjoy--Carleman and Braun--Meise--Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.