Relationship Between the Bojanov–Naidenov Problem and the Kolmogorov-Type Inequalities

Abstract

It is shown that the Bojanov–Naidenov problem \({\Vert {x}^{\left(k\right)}\Vert }_{q, \delta }\) → sup, k = 0, 1, . . . , r − 1, on the classes of functions \({\Omega }_{p}^{r}\left({A}_{0}, {A}_{r}\right)\) := \(\left\{x \in {L}_{\infty }^{r}: {\Vert {x}^{\left(r\right)}\Vert }_{\infty }\le {A}_{r}, L{\left(x\right)}_{p}\le {A}_{0}\right\},\) where q ≥ 1 for k ≥ 1 and q ≥ p for k = 0, is equivalent to the problem of finding the sharp constant C = C(λ) in the Kolmogorov-type inequality

\({\Vert {x}^{\left(r\right)}\Vert }_{q,\delta }\le CL{\left(x\right)}_{p}^{\alpha }{\Vert {x}^{\left(r\right)}\Vert }_{\infty }^{1-\alpha }, x\in {\Omega }_{p,\lambda }^{r}, (1)\)

where \(\alpha =\frac{r-k+1/q}{r+1/p},\) \({\Vert x\Vert }_{p,\delta }\) := sup {\({\Vert x\Vert }_{{L}_{p}[a,b]}\):a, b, ∈ R, 0 < b – a ≤ δ} δ > 0, \({\Omega }_{p,\lambda }^{r}\) := \(\bigcup \left\{{\Omega }_{p}^{r}\left({A}_{0}, {A}_{r}\right):{A}_{0}={A}_{r}L\left(\varphi \lambda ,r\right)p\right\},\) ⋋ > 0, φ⋋,r is a contraction of the ideal Euler spline of order r, and L_(x)p : = sup {\({\Vert x\Vert }_{{L}_{p}[a,b]}:\) a, b, ∈ R |x(t)| > 0, t ∈ (a,b)}. In particular, we obtain a sharp inequality of the form (1) in the classes \({\Omega }_{p,\lambda }^{r},\) ⋋ > 0. We also prove the theorems on relationships for the Bojanov–Naidenov problems in the spaces of trigonometric polynomials and splines and establish the corresponding sharp Bernstein-type inequalities.