The main result is a generalization of N. P. Kornejchuk’s lemma [“Extremal problems in approximation theory” (Russian Nauka, Moskow 1976), Sect. 7.4] about extremal functions in the problem \(\int_{a}^{b}h(t)\psi(t)dt\to \sup\), \(h\in H^{\omega }[a,b]\). Namely, the author finds a fine generalization of rearrangements of simple kernels being used by N. P. Kornejchuk to kernels with finitely many points of sign change on \([a,b].\)
Applications to various versions of the Kolmogorov-Landau problem \[ f^{(m)}(0)\to \sup , \quad f\in W^{r}H^{\omega }(\mathbb I), \quad \| f \| _{L_{\infty }(\mathbb I)}\leq B, \] where \(0<m\leq r\), \(B>0, \mathbb I = \mathbb R _{+},\) or \([0,1],\) are discussed.
The exposition is clear and nicely constructed. A highly recommended paper for everybody interested in solutions of different extremal problems of approximation theory.