Boyanov–Naidenov problem and the Kolmogorov-type inequalities for functions on the real axis

Abstract

UDC 517.5

We solve the Boyanov–Naidenov problem $\big\|x^{(k)}\big\|_{q,\, \delta} \to \sup,$ $k= 1,\ldots ,r-1,$ $q \ge 1,$ on the classes of functions $W^r_{p,\varepsilon}(A_0, A_r):=\big\{x\in L^r_{\infty}\colon \|x\|_{p, \varepsilon} \le A_0 ,\ \big\|x^{(r)}\big\|_{\infty} \le A_r \big\},$ where $\|x\|_{p, \delta}:=\sup \big\{ \|x\|_{L_p[a,\, b]}\colon a, b \in {\rm \bf R}, \ 0< b-a \le \delta \big\},$ $p, \delta > 0,$ $\varepsilon \in (0, \varepsilon_1 ],$ $\varepsilon_1 := \pi / \omega,$ the number $\omega$ satisfies the condition $A_0 = A_r \|\varphi_{\omega, r}\|_{p,\, \pi / \omega},$  $\varphi_{\omega, r}(t):=\omega^{-r}\varphi_{ r}(\omega t),$ and $\varphi_{ r}$ is the ideal Euler spline of order $r.$ In addition, we prove that the Boyanov–Naidenov problem is equivalent to the problem of sharp constant $C = C(\lambda)$ in the Kolmogorov-type inequality \begin{gather}\big\|x^{(k)}\big\|_{q,\, \delta} \leq C \|x\|_{p,\, \varepsilon}^{\alpha} \big\|x^{(r) }\big\|_\infty^{1-\alpha},  \quad x\in L^{r, \lambda}_{p,\varepsilon}, \tag{1}  \end{gather} where $\alpha=\dfrac{r-k+1/q}{r+1/p},$ $L^{r, \lambda}_{p,\varepsilon}:= \big\{x\in L^r_{\infty}\colon \|x\|_{p,\, \varepsilon} = \|\varphi_{\lambda, r}\|_{p,\, \varepsilon}\cdot \big\|x^{(r)}\big\|_{\infty} \big\},$ and $\lambda > 0.$ In particular, we obtain the sharp inequality of the form (1).

We also solve the Boyanov–Naidenov problem in the spaces of trigonometric polynomials and splines and establish the theorems on the relationship between this problem and sharp inequalities of the Bernstein type.  As a consequence, we prove sharp inequalities of the indicated  type for polynomials and splines.