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Nonlinear Conditions for Ultradifferentiability.

David Nicolas Nenning1, Armin Rainer1, Gerhard Schindl1

  • 1Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.

Journal of Geometric Analysis
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

Joris's theorem on function properties is extended to ultradifferentiable classes. This division property holds broadly for non-quasianalytic functions and in specific quasianalytic cases, expanding the scope of this mathematical result.

Keywords:
(Non-)QuasianalyticAlmost analytic extensionDivision propertyHolomorphic approximationJoris theoremUltradifferentiable classes

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Area of Science:

  • Real Analysis
  • Harmonic Analysis
  • Function Theory

Background:

  • Joris's theorem provides conditions for function properties based on powers.
  • Thilliez extended this to Roumieu-type Denjoy-Carleman classes.
  • Ultradifferentiable classes are crucial in advanced analysis.

Purpose of the Study:

  • To investigate the general validity of Joris's division property in various ultradifferentiable classes.
  • To determine the conditions under which this property holds across different dimensions and types of classes (quasianalytic vs. non-quasianalytic).

Main Methods:

  • Analysis of function properties within the framework of ultradifferentiable classes.
  • Application of techniques from real and harmonic analysis.
  • Comparative study of quasianalytic and non-quasianalytic settings.

Main Results:

  • The division property, equivalent to Joris's theorem, is proven to be valid in a wide range of ultradifferentiable classes.
  • This property holds in all dimensions for non-quasianalytic classes.
  • In the quasianalytic case, validity is established in dimension one and for specific classes in all dimensions.

Conclusions:

  • The study generalizes Joris's theorem to a broad spectrum of ultradifferentiable function classes.
  • The findings highlight the behavior of this division property in both quasianalytic and non-quasianalytic settings.
  • This work contributes to a deeper understanding of function theory in generalized classes.