Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

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.