The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

Abstract

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 $$$q \geqslant 1$$$.