On the Norms of Boman–Shapiro Difference Operators

Abstract

For given k ∈ N and h > 0, an exact inequality kW2k(f, h)kC ≤ Ck kfkC is considered on the space C = C(R) of continuous functions bounded on the real axis R = (−∞, ∞) for the Boman–Shapiro difference operator W2k(f, h)(x):= (−h1)k Z−hh (2kk)−1 ∆b 2tkf(x) ( 1 − |ht| ) dt, where ∆b 2tkf(x):= P (−1)j(2jk)f(x + jt − kt) is the 2k j=0 central finite difference of a function f of order 2k with step t. For each fixed k ∈ N, the exact constant Ck in the above inequality is the norm of the operator W2k(·, h) from C to C. It is proved that Ck is independent of h and increases in k. A simple method is proposed for the calculation of the constant C∗ = limk→∞ Ck = 2.6699263 . . . with accuracy 10−7. We also consider the problem of extending a continuous function f from the interval [−1, 1] to the axis R. For extensions gf := gf,k,h, k ∈ N, 0 < h < 1/(2k), of functions f ∈ C[−1, 1], we obtain new two-sided estimates for the exact constant Ck∗ in the inequality kW2k(gf, h)kC(R) ≤ Ck∗ ω2k(f, h), where ω2k(f, h) is the modulus of continuity of f of order 2k. Specifically, for any natural k ≥ 6 and any h ∈ (0, 1/(2k)), we prove the double inequality 5/12 ≤ Ck∗ < (2 + e−2) C∗. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.