Classification of infinitely differentiable periodic functions

The set of infinitely differentiable periodic functions is studied in terms of generalized -derivatives defined by a pair of sequences ψ 1 and ψ 2 . In particular, we establish that every function f from the set has at least one derivative whose parameters ψ 1 and ψ 2 decrease faster than any power...

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Bibliographic Details
Published in:Ukrainian mathematical journal 2008-12, Vol.60 (12), p.1982-2005
Main Authors: Stepanets, A. I., Serdyuk, A. S., Shidlich, A. L.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Geometry
Mathematics
Mathematics and Statistics
Statistics
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Summary:The set of infinitely differentiable periodic functions is studied in terms of generalized -derivatives defined by a pair of sequences ψ 1 and ψ 2 . In particular, we establish that every function f from the set 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 ∈ 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 -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.
ISSN:0041-5995
1573-9376
DOI:10.1007/s11253-009-0185-1