From Ultra-differentiable to Quasi-Analytic Analysis, Lecture 4

Abstract: The mini-course is intended as an introduction to ultradifferential analysis with special emphasis on ultra-differentiable extension theorems. The development of differential analysis in the last century was decisively influenced by Whitney’s work on the extension of differentiable functions from closed sets. We shall be interested in quantitative versions of Whitney’s extension theorem. The quantitative aspect is implemented by uniform growth properties of the multisequence of partial derivatives which measure the deviation from the Cauchy estimates and hence from analyticity. These growth conditions determine ultradifferentiable function classes which form a scale of regularity classes between the real analytic and the smooth class.

Lecture 4. An ultradifferentiable Whitney approximation theorem will enable us to conclude that the extension can always be chosen real analytic off the given closed set. Furthermore, we will discuss the extension problem with a controlled loss of regularity. This requires a different approach which is closely related to the characterization of ultradifferentiable classes by almost analytic extensions. We will address several applications such as Lojasiewicz’s theorem on regularly situated sets or Whitney’s spectral theorem in the ultradifferentiable framework. Finally, we shall take a glimpse on other ultradifferentiable classes, in particular, Braun–Meise–Taylor classes.

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Ultra-differentiable Analysis concerns sub-algebras of smooth functions with constrained growth of the Taylor series coefficients. Besides its importance in the analysis of partial differential equations the development of this theory was influenced by the classical Whitney extension problem and the composition problem. Both problems followed a path that often meets sub-analytic geometry, and later o-minimal geometry. Of special interest among the ultra-differentiable classes are the quasi-analytic classes which possess a (quasi-)analytic continuation property similar to the real analytic class. This property make these classes interesting from an analytic viewpoint and also suitable for questions of tame geometry. In relation with asymptotic expansions of solutions of analytic ODEs they also naturally appear when dealing with non-oscillation problems of the solutions. Armin Rainer will introduce us to ultra-differentiable classes, in particular to quasi-analytic ones.

algebraic geometrydifferential geometrymetric geometry

Audience: advanced learners

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School of Real Geometry in Fortaleza - ScReGeFor

Series comments: Main aim of School of Real Geometry in Fortaleza is to present current topics in Real Algebraic and Tame Geometries. There will be five courses on topics in Real Geometry. The target audience is primarily Phd Students, Post-docs and young researchers.

All courses and talks will be available online in google meet. The links will be sent to registered participants some days before the event.

More information available on the website sites.google.com/view/scregefor2020/home

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