Inequalities of Bernstein type for splines in $L_2(\mathbb{R})$ space

We obtain sharp inequality of Bernstein type in $L_2(\mathbb{R})$ space for non-periodic spline functions of degree $m$, of minimal defect, with equidistant knots.

The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

For given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $r$ and minimal defect with knots at the points $k\pi / n$, $k \in \mathbb{Z}$, such that $\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$, $\delta \in (0, \pi / n]$, where $\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$ and $\varphi_{n, r}$ is the $(2\pi / n)$-periodic spline of Euler of order $r$. In particular, we solve the same problem for the intermediate derivatives $x^{(k)}$, $k = 1, ..., r-1$, with $q \geqslant 1$.

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

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 $[a, b] \subset \mathbb{R}$.