The problem of the asymmetric ideal spline least deviating from zero in the \(C[a,b]\)-metric is solved. The authors prove the Landau-Kolmogorov-Hörmander inequalities for the norms of positive and negative parts of intermediate derivatives of functions on the semiaxis that take into account restrictions on the positive and negative part of the higher derivative. The well-known Schönberg-Cavaretta inequality is generalized and refined.