The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative
Abstract
For given r∈\NN, p,α,β,μ>0, we solve the
extremal problems
∫baxq±(t)dt→sup,q≥p,
on the set of pair (x,I) functions x∈Lr∞ and
intervals I=[a,b]⊂\RR satisfying the inequalities −β≤x(r)(t)≤α for almost everywhere t∈\RR and
the both of conditions L(x±)p≤L(φα,βλ,r)±)p, and such that μ(supp[a,b]x+)≤μ or μ(supp[a,b]x−)≤μ, where
L(x)p:=sup{‖
{\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\}
and \varphi_{\lambda,r}^{\alpha, \beta} is the nonsymmetric
(2\pi/\lambda)-periodic spline of Euler of order r. In
particular, we solve the same problems for the intermediate
derivatives x^{(k)}_{\pm}, k=1,...,r-1, with q \ge 1.
Copyright (c) 20219 Володимир Олександрович Кофанов
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