Bojanov–Naidenov problem for differentiable functions and the Erdös problem for polynomials and splines - Kofanov, V. (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 517.5We solve the extremal problem $$\big\|x^{(k)}_{±}\big\|_{L_p[a, b]} → \sup, \quad k=0,1,… ,r-1,\quad p > 0 ,$$ in the class of pairs $(x, I)$ of functions $x\in S^k_{φ}$ such that $ φ^{(i)} $ are the comparison functions for $ x^{(i)},$ $i=0, 1,… ,k,$ and the intervals $I = [a,b]$ satisfy the conditions $$L(x)_p \le A,\quad μ \big\{{\rm supp}_{[a, b]} x^{(k)}_{±}\big\} \le μ ,$$ where $L(x)_p:=\sup \left\{\left( \displaystyle \int olimits_{a}^{b}|x (t)| ^pdt\right) ^{\!\frac1p}\colon a, b \in {\rm \bf R},\ |x(t)|>0,\ t\in (a, b) \right\}.$ In particular, we solve the same problems on the classes $W^r_∞({\rm \bf R})$ and on the bounded sets of spaces of trigonometric polynomials and splines and the Erdös problem for the positive (negative) parts of polynomials and splines.