Inverse theorems on the approximation of periodic functions with high generalized smoothness (Q2113438)

The authors proved the validity of the inverse estimate of Bernstein type \[ \left\Vert f^{\left( \lambda ,\beta \right) }\right\Vert _{p}\leq c\sum_{k=0}^{\infty }\left( \lambda _{k+1}-\lambda _{k}\right) E_{k}\left(f\right) _{p},\text{ }f\in L^{p}\text{ }\left( p\geq 1\right) \] and of its direct corollaries \[ E_{n-1}\left( f^{\left( \lambda ,\beta \right) }\right) _{p}\leq c\left(\lambda _{n}E_{n-1}\left( f\right) _{p}+\sum_{k=n}^{\infty }\left( \lambda _{k+1}-\lambda _{k}\right) E_{k}\left( f\right) _{p}\right) \] \[ \left\Vert T^{\left( \lambda ,\beta \right) }\right\Vert _{p}\leq c\lambda_{n}\left\Vert T\right\Vert _{p},\text{ }c\in\mathbb{C},\text{ }\lambda _{n}\nearrow \infty \text{, }\beta \in\mathbb{R}, \] where \(E_{n}\left( f\right) _{p}\) is the best approximation of a function \(f\in L^{p}\) by trigonometric polynomials \(T\) of degree at most \(n\in \mathbb{N}\) and \[ \left( \cdot \right) ^{\left( \lambda ,\beta \right) }:e^{ikx}\rightarrow\lambda _{k}e^{sgnk\left( i\pi \beta \right) /2}e^{ikx},\text{ }k\in \mathbb{Z}, \] is the linear operator of the \(\left( \lambda ,\beta \right) \)-derivative defined on the corresponding class of smooth functions in the generalized sense (as distributions).