An interpolation problem in the Denjoy–Carleman classes

Paolo Albano and Marco Mughetti

Mathematische Nachrichten, 2023, vol. 296, issue 3, 902-914

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]⊂R$[a,b]\subset {\mathbb {R}}$ and given an increasing divergent sequence dn$d_n$ of positive integers such that the derivative of order dn$d_n$ of f has a growth of the type Mdn$M_{d_n}$, when can we deduce that f is a function in the Denjoy–Carleman class CM([a,b])$C^M([a,b])$? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence dn$d_n$ is needed.

Date: 2023
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