An interpolation problem in the Denjoy-Carleman classes

TL;DR

This paper investigates conditions under which a smooth function belongs to a Denjoy-Carleman class based on the growth of selected derivatives, providing new criteria linked to the sequence of derivative orders.

Contribution

It introduces a novel criterion involving the gaps in the derivative order sequence to determine membership in Denjoy-Carleman classes.

Findings

A positive criterion for membership in Denjoy-Carleman classes is established.

A necessary condition on the gaps between derivative orders is identified.

The results extend understanding of interpolation in Denjoy-Carleman classes.

Abstract

Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on $[a,b]\subset\mathbb{R}$ and given an increasing divergent sequence $d_n$ of positive integers such that the derivative of order $d_n$ of $f$ has a growth of the type $M_{d_n}$, when can we deduce that $f$ is a function in the Denjoy-Carleman class $C^M([a,b])$? We provide a positive result, and we show that a suitable condition on the gaps between the terms of the sequence $d_n$ is needed.