Bojanov–Naidenov problem for differentiable functions and the Erdös problem for polynomials and splines

• Datum: 2023
• Hauptverfasser: Kofanov, V., Кофанов, Владимир Александрович, Кофанов, Володимир
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2023
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/7259
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 We solve the extremal problem $$\big\|x^{(k)}_{\pm}\big\|_{L_p[a, b]} \to \sup, \quad k=0,1,\ldots ,r-1,\quad p > 0 ,$$ in the class of pairs $(x, I)$ of functions $x\in S^k_{\varphi}$ such that $ \varphi^{(i)} $ are the comparison functions for  $ x^{(i)},$ $i=0, 1,\ldots ,k,$ and the intervals $I = [a,b]$ satisfy the conditions $$L(x)_p \le A,\quad \mu \big\{{\rm supp}_{[a, b]} x^{(k)}_{\pm}\big\} \le \mu ,$$ where $L(x)_p:=\sup \left\{\left( \displaystyle \int\nolimits_{a}^{b}|x (t)| ^pdt\right) ^{\!\frac1p}\colon  a, b \in {\rm \bf R},\ |x(t)|>0,\ t\in (a, b) \right\}.$ In particular, we solve the same problems on the classes $W^r_\infty({\rm \bf R})$ and on the bounded sets of spaces of trigonometric polynomials and splines and the Erd\"{o}s problem for the positive (negative) parts of polynomials and splines.