Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives
We solve the following extremal problems: (i) \( s^(k)_L_q[α, β ]→ \) and (ii) \( s^(k)_W_q→ \) 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:= {x_L_p[a,b]:a,bR,0.5em |x(t)|>0,0.5em t(a,b)\