Sharp Bernstein Type Inequalities for Splines in the Mean Square Metrics

We give an elementary proof of the sharp Bernstein type inequality

$$ {\left\Vert {f}^{(s)}\right\Vert}_2\le \frac{n^s}{2^s}{\left(\frac{\kappa_{2r+1-2s}}{\kappa_{2r+1}}\right)}^{1/2}{\left\Vert {\updelta}_{\frac{\uppi}{n}}^sf\right\Vert}_2. $$

Here n, r, s ∈ ℕ, f is a 2π-periodic spline of order r and of minimal defect with nodes $\frac{\mathrm{j}\uppi}{n}$ , j ∈ Z, δ ^s _h is the difference operator of order s with step h, and the K _m are the Favard constants. A similar inequality for the space L ₂(ℝ) is also established. Bibliography: 5 titles.