On exact Bernstein-type inequalities for splines - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines s of order r and defect 1 with nodes at the points iπ/n, i ∈ Z, n ∈ N: \| s^(k) \|_q n^k + 1/p - 1/q \| φ _r - k \|_q \| φ _r \|_p \| s \|_p , where k, r ∈ N, k < r, p = 1 or p = 2, q > p, and ϕr is the perfect Euler spline of order r.