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On real analytic functions on closed subanalytic domains.

Armin Rainer1

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

Archiv Der Mathematik. Archives of Mathematics. Archives Mathematiques
|May 14, 2024
PubMed
Summary

Real analytic functions on specific sets are identified by their smoothness and analytic composites with polynomial curves. This holds even for non-smooth functions if their composites with quadratic polynomial maps are analytic.

Keywords:
Real analyticity on closed sets, Cuspidality of sets, Bochnak–Siciak theorem, Subanalytic sets, Uniformly polynomially cuspidal sets

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Area of Science:

  • Real analytic functions
  • Complex analysis
  • Geometric measure theory

Background:

  • Characterizing real analytic functions on complex sets is a fundamental problem.
  • Polynomially cuspidal sets present unique challenges due to their intricate geometric structure.

Purpose of the Study:

  • To establish criteria for determining if a function is real analytic on a closed uniformly polynomially cuspidal set.
  • To investigate the role of polynomial curves and their composites in this characterization.

Main Methods:

  • Analysis of function composition with polynomial curves.
  • Utilizing properties of uniformly polynomially cuspidal sets.
  • Investigating the impact of boundary regularity (e.g., Lipschitz continuity).

Main Results:

  • A function is real analytic if and only if it's smooth and its composites with polynomial curves are real analytic.
  • The degree of polynomial curves depends on the boundary regularity of the set.
  • For Lipschitz boundaries, quadratic polynomial maps suffice, even without assuming initial smoothness.

Conclusions:

  • The study provides a novel characterization of real analytic functions on polynomially cuspidal sets.
  • The regularity of the set's boundary is crucial for the degree of polynomial curves required.
  • The findings offer new perspectives on analytic extension and function properties on complex geometric structures.