On the Norms of Boman–Shapiro Difference Operators

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 $W_{2k}(f,h)(x):=\displaystyle\frac{(-1)^k}{h}\displaystyle\int\nolimits_{-h}^h\!{\binom {2k} k}^{\!-1}\widehat \Delta_t^{2k}f(x)\Big(1-\frac{|t|}h\Big)\, dt$, where $\widehat\Delta_t^{2k} f(x):=\sum\nolimits_{j=0}^{2k} (-1)^{j} \binom{2k}{j} f(x+jt-kt)$ 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 $C_{*}=\lim_{k\to\infty}C_{k}=2.6699263\dots$ 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 every positive integer $k\ge 6$ and every $h\in\big(0,1/(2k)\big)$, we prove the double inequality $5/12\le C^{*}_{k}<\big(2+e^{-2}\big)\,C_{*}$.