Multiple modules of continuity and the best approximations of periodic functions in metric spaces - Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
It is proved that, under the condition $M_{Ψ} \Bigl( \frac 12\Bigr) < 1$, where $M_{Ψ}$ is a stretching function $Ψ$ in the space $L_{Ψ}$ , the Jacksoninequalities$$\sup_n \sup_{f\in L_{Ψ}, f ot = \text{const}} \frac{E_{n-1}(f)_{Ψ} }{ω_k \Bigl(f, \frac{π}n\Bigr)_{Ψ}} < ∞,$$are true; here, $E_{n-1}(f)_{Ψ}$ is the best approximation of $f$ by trigonometric polynomials of degree at most $n - 1$ and $ω_k \Bigl(f, \frac{π}n \Bigr)_{Ψ}$ is the modulus of continuity of $f$ of order $k$, $k \in N$. We study necessary and sufficient conditions for thefunction $f$ under which the following relation is true: $E_{n-1}(f)_{Ψ} \asymp ω_k\Bigl(f, \frac{π}n\Bigr)_{Ψ}.$