On some new criteria for infinite differentiability of periodic functions

Abstract

We study the set $\mathcal{D}^\infty$ of infinitely differentiable periodic functions in terms of generalized $\bar \psi$-derivatives defined by a pair $\bar \psi = (\psi _1 ,\psi _2 )$ of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set $\mathcal{D}^\infty$ has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ $\mathcal{D}^\infty$ 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 $\bar \psi$-derivative no longer exists.