Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

  • V. A. Kofanov

Abstract

We solve the following extremal problems: (i) \( {\left\Vert {s}^{(k)}\right\Vert}_{L_q\left[\alpha, \beta \right]}\to \sup \) and (ii) \( {\left\Vert {s}^{(k)}\right\Vert}_{W_q}\to \sup \) 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, $$ L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \mathbf{R},\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\} $$

and \( {\left\Vert \cdot \right\Vert}_{W_q} \) is the Weyl functional, i.e., $$ {\left\Vert x\right\Vert}_{W_q}:=\underset{\varDelta \to \infty }{ \lim}\underset{a\in \mathbf{R}}{ \sup }{\left(\frac{1}{\varDelta }{\displaystyle \underset{a}{\overset{a+\varDelta }{\int }}{\left|x(t)\right|}^qdt}\right)}^{1/q}. $$

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

Published
25.02.2014
How to Cite
KofanovV. 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, http://umj.imath.kiev.ua/index.php/umj/article/view/2125.
Section
Research articles