# Classification of infinitely differentiable periodic functions

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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## Preview

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## References

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