Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

Abstract

We solve the following extremal problems: (i) ks (k) k Lq [↵,β] ! sup and (ii) ks (k) kW q ! sup over all shifts of splines of order r with minimal defect and nodes at the points lh, l 2 Z, such that L(s)p  M in the cases: , β] is an arbitrary interval in the real line,o and k · kW q is the Weyl functional, i.e.,As a special case, we get some generalizations of the Ligun inequality for splines.