Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives
Abstract
We solve the following extremal problems: (i) ‖s(k)‖Lq[α,β]→sup and (ii) ‖s(k)‖Wq→sup over all shifts of splines of order r with minimal defect and nodes at the points lh, l ∈ Z , such that L(s) p ≤M in the cases: (a) k =0, q ≥ p >0, (b) k =1, . . . , r −1, q ≥ 1, where [α, β] is an arbitrary interval in the real line, L(x)p:=sup{‖x‖Lp[a,b]:a,b∈R,|x(t)|>0,t∈(a,b)}and ‖⋅‖Wq is the Weyl functional, i.e., ‖x‖Wq:=limΔ→∞supa∈R(1Δa+Δ∫a|x(t)|qdt)1/q.
As a special case, we get some generalizations of the Ligun inequality for splines.
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Published
25.02.2014
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Research articles
How to Cite
Kofanov, V. A. “Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives”. Ukrains’kyi Matematychnyi Zhurnal, vol. 66, no. 2, Feb. 2014, pp. 216–225, https://umj.imath.kiev.ua/index.php/umj/article/view/2125.