Extremal problems for non-periodic splines on real domain and their derivatives

Abstract

We consider the Bojanov-Naidenov problem over the set $$$\sigma_{h,r}$$$ of all non-periodic splines $$$s$$$ of order $$$r$$$ and minimal defect with knots at the points $$$kh$$$, $$$k \in \mathbb{Z}$$$. More exactly, for given $$$n, r \in \mathbb{N}$$$; $$$p, A > 0$$$ and any fixed interval $$$[a, b] \subset \mathbb{R}$$$ we solve the following extremal problem $$$\int\limits_a^b |x(t)|^q dt \rightarrow \sup$$$, $$$q \geqslant p$$$, over the classes $$$\sigma_{h,r}^p(A) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \delta \in (0, h], \tau \in \mathbb{R} \bigr\}$$$, where $$$\| x \|_{p, \delta} := \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a \leqslant \delta \bigr\}$$$, and $$$\varphi_{\lambda, r}$$$ is $$$(2\pi / \lambda)$$$-periodic spline of Euler of order $$$r$$$. In particularly, for $$$k = 1, ..., r - 1$$$ we solve the extremal problem $$$\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$$$, $$$q \geqslant 1$$$, over the classes $$$\sigma_{h,r}^p (A)$$$. Note that the problems (1) and (2) were solved earlier on the classes $$$\sigma_{h,r}(A, p) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \tau \in \mathbb{R} \bigr\}$$$, where $$$L(x)_p := \sup \bigl\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \bigr\}$$$. We prove that the classes $$$\sigma_{h,r}^p (A)$$$ are wider than the classes $$$\sigma_{h,r}(A,p)$$$. Similarly we solve the analog of Erdös problem about the characterisation of the spline $$$s \in \sigma_{h,r}^p(A)$$$ that has maximal arc length over fixed interval $$$[a, b] \subset \mathbb{R}$$$.