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Title: On the Norms of Boman–Shapiro Difference Operators
Other Titles: О нормах разностных операторов Бомана — Шапиро
Authors: Babenko, A. G.
Kryakin, Yu. V.
Issue Date: 2020
Publisher: Krasovskii Institute of Mathematics and Mechanics
Krasovskii Institute of Mathematics and Mechanics UB RAS
Citation: Babenko A. G. On the Norms of Boman–Shapiro Difference Operators [О нормах разностных операторов Бомана — Шапиро] / A. G. Babenko, Yu. V. Kryakin // Trudy Instituta Matematiki i Mekhaniki UrO RAN. — 2020. — Vol. 26. — Iss. 4. — P. 64-75.
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.
Access: info:eu-repo/semantics/openAccess
RSCI ID: 44314659
SCOPUS ID: 85103627058
PURE ID: 20232940
ISSN: 0134-4889
DOI: 10.21538/0134-4889-2020-26-4-64-75
metadata.dc.description.sponsorship: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University), and as part of research conducted in the Ural Mathematical Center.
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