On Taylor coefficients of smooth functions

TL;DR

This paper characterizes the image of the Borel map for smooth functions, revealing a threshold at the logarithmic class and establishing a duality between quasianalytic and non-quasianalytic classes, with implications for recovering functions from Taylor coefficients.

Contribution

It provides a complete description of the Borel map's image for Beurling classes and introduces a moment-type summation method for function recovery from Taylor jets.

Findings

Threshold at the logarithmic class in the image description

Duality between non-quasianalytic and quasianalytic classes

Classical results of Carleson and Ehrenpreis are complemented

Abstract

We study the Borel map, which maps infinitely differentiable functions on an interval to the jets of their Taylor coefficients at a given point in the interval. Our main results include a complete description of the image of the Borel map for Beurling classes of smooth functions and a moment-type summation method which allows one to recover a function from its Taylor jet. A surprising feature of this description is an unexpected threshold at the logarithmic class. Another interesting finding is a "duality" between non-quasianalytic and quasianalytic classes, which reduces the description of the image of the Borel map for non-quasianalytic classes to the one for the corresponding quasianalytic classes, and complements classical results of Carleson and Ehrenpreis.