Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

Authors

  • V. A. Kofanov

Abstract

We solve the following extremal problems: (i) s(k)Lq[α,β]sup and (ii) s(k)Wqsup 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{xLp[a,b]:a,bR,|x(t)|>0,t(a,b)}

and Wq is the Weyl functional, i.e., xWq:=limΔsupaR(1Δa+Δa|x(t)|qdt)1/q.

As a special case, we get some generalizations of the Ligun inequality for splines.

Published

25.02.2014

Issue

Section

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.