Inequalities of bernstein type for polynomial splines in L₂

Abstract

For 2ir-periodic polynomial splines of order r, of minimal defect, with nodes at the points ktr/n, ne, there are established the sharp inequalities

$$\parallel s^{(1)} \parallel _{ 2} \leqslant \frac{{\parallel \varphi _{n,r}^{(1)} \parallel _{ 2} }}{{\parallel \Delta _h^l \varphi _{n, r} \parallel _{ 2} }}\parallel \Delta _h^l s\parallel _{ 2} \leqslant \frac{{\parallel \varphi _{n,r}^{(1)} \parallel _{ 2} }}{{\parallel \varphi _{n, r} \parallel _{ 2} }}\parallel s \parallel _{ 2} , l = 1, ..., r - 1,$$

valid for 0<h<ir/2n and 0<­<ir/4n respectively, where <i,.r is the r-th periodic integral of the function (*)=sign sin nx, and

$$\Delta _h^l / (x) = \sum\limits_{k = 0}^l {( - 1)^k C_l^k /} (x + (l - 2k) h)$$

.