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Ukrainian Mathematical Journal

, Volume 60, Issue 12, pp 1982–2005 | Cite as

Classification of infinitely differentiable periodic functions

  • A. I. Stepanets
  • A. S. Serdyuk
  • A. L. Shidlich
Article

The set \( \mathcal{D}^\infty \) of infinitely differentiable periodic functions is studied in terms of generalized \( \overline \psi \)-derivatives defined by a pair \( \overline \psi = (\psi_1, \psi_2)\) of sequences ψ 1 and ψ 2. In particular, we establish that every function f from the set \( \mathcal{D}^\infty \) has at least one derivative whose parameters ψ 1 and ψ 2 decrease faster than any power function. At the same time, for an arbitrary function f\( \mathcal{D}^\infty \) different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ 1 and ψ 2 have the same rate of decrease and for which the \( \overline \psi \)-derivative no longer exists. We also obtain new criteria for 2π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions.

Keywords

Fourier Series Entire Function Real Axis Trigonometric Polynomial Tangency Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • A. I. Stepanets
    • 1
  • A. S. Serdyuk
    • 1
  • A. L. Shidlich
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

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