Nikolskii type inequality for cardinal splines

Abstract

The Nikolskii type inequality for cardinal splines

$$||s||_{p(R)} \leqslant 2(1 + 4\pi )2^{1 - \tfrac{q}{p}} \left( {\frac{\pi }{h}} \right)^{\tfrac{1}{q} - \tfrac{1}{p}} ||s||_{ q(R)} ,0 < q < p \leqslant \infty $$

is proved, which is exact in the sense of order, where ∈ ℒ _m,h , and ℒ _m,k is the space of cardinal splines with nodes

$$\left\{ {jh + \frac{1}{2}(m - 1)h} \right\}_{j \in z} $$