A.G. Babenko, Yu.V. Kryakin. On the norms of Boman–Shapiro difference operators ... P. 64-75

For given $k\in\mathbb N$ and $h>0$, an exact inequality $\|W_{2k}(f,h)\|_{C}\le C_{k}\,\|f\|_{C}$ is considered on the space $C=C(\mathbb R)$ of continuous functions bounded on the real axis $\mathbb R=(-\infty,\infty)$ for the Boman-Shapiro difference operator $$W_{2k}(f,h)(x):=\displaystyle\frac{(-1)^k}{h}\displaystyle\int\limits_{-h}^h\!{\binom {2k} k}^{\!-1}\widehat \Delta_t^{2k}f(x)\Big(1-\frac{|t|}h\Big)\, dt,$$ where $\widehat\Delta_t^{2k} f(x):=\sum_{j=0}^{2k} (-1)^{j} \binom{2k}{j} f(x+jt-kt)$ is the central finite difference of a function $f$ of order $2k$ with step $t$. For each fixed $k\in\mathbb N$, the exact constant $C_{k}$ in the above inequality is the norm of the operator $W_{2k}(\cdot,h)$ from $C$ to $C$. It is proved that $C_{k}$ is independent of $h$ and increases in $k$. A simple method is proposed for the calculation of the constant $C_{*}=\lim\limits_{k\to\infty}C_{k}=2.6699263\dots$ with accuracy $10^{-7}$. We also consider the problem of extending a continuous function $f$ from the interval $[-1,1]$ to the axis $\mathbb{R}$. For extensions $g_f:=g_{f,k,h}$, $k\in\mathbb N$, $0<h<1/(2k)$, of functions $f\in C[-1,1]$, we obtain new two-sided estimates for the exact constant $C^{*}_{k}$ in the inequality $\|W_{2k}(g_f,h)\|_{C(\mathbb R)}\le C^{*}_{k}\,\omega_{2k}(f,h)$, where $\omega_{2k}(f,h)$ is the modulus of continuity of $f$ of order $2k$. Specifically, for any natural $k\ge 6$ and any $h\in\big(0,1/(2k)\big)$, we prove the double inequality $5/12\le C^{*}_{k}<\big(2+e^{-2}\big) \,C_{*}$.

Keywords: difference operator, $k$th modulus of continuity, norm estimate

Received July 13, 2020

Revised November 15, 2020

Accepted November 23, 2020

Funding Agency: 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.

Aleksandr Grigor’evich Babenko, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: babenko@imm.uran.ru

Yuriy Kryakin, dr hab., Mathematical Institute of University of Wroclaw, 48-300 Wroclaw, Poland, e-mail: kryakin@math.uni.wroc.pl

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Cite this article as: A.G. Babenko, Yu.V. Kryakin. On the norms of Boman–Shapiro difference operators, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 64–75; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 315, Suppl. 1, pp. S55–S66.