Exact inequalities for splines and best quadrature formulas for certain classes of functions

In this note inequalities between the norms of a spline and its derivatives in various Orlich spaces are obtained. These inequalities are analogs of the inequalities of L. V. Takov for trigonometrical polynomials and generalize S. N. Bernstein's inequalities. An inequality for monosplines which reduces to the best quadrature formula for the classes $W^rL_1$, where $r=1,2,\dots$, is also obtained. For $r=2,4,6,\dots$ this result was obtained earlier by V. P. Motornyi.