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Local saturation of conservative operators. (English) Zbl 1058.41017

The authors state a result about the local saturation of sequences of linear operators that preserve the sign of $k$-th derivative of the functions.

Let $I\subset ℝ$, $i\in {ℕ}_{0}=ℕ\cup \left\{0\right\}$. Let ${C}^{i}\left(I\right)$ be the space of all real-valued and $i$-times continuously differentiable functions defined on $I$ and ${D}^{i}$ be the $i$-th differential operator with $C\left(I\right)={C}^{0}\left(I\right)$ the space of all continuous functions defined on $I$ and ${D}^{0}$ the identity operator, respectively. Besides, ${C}_{B}^{i}\left(I\right)$ denotes the subspace formed by the functions of ${C}^{i}\left(I\right)$ which are bounded on $I$. A function $f\in {C}^{i}\left(I\right)$ is said to be $i$-convex if ${D}^{i}f\ge 0$ on $I$ and a linear operator is said to be $i$-convex if it maps $i$-convex functions onto $i$-convex functions.

Definition. Let $L\left(y\right)=0$ denote the second-order linear differential equation $L\left(y\right)\equiv {D}^{2}y+{p}_{1}\left(t\right){D}^{1}y+{p}_{2}\left(t\right)y=0$. Consider an interval $\left(a,b\right)$ such that $L\left(y\right)=0$ has a unique solution, continuous in $\left(a,b\right)$, taking any given real values ${y}_{1}$ and ${y}_{2}$ at any two given ${t}_{1}$ and ${t}_{2}$ within $\left(a,b\right)$. Assume that ${p}_{1}\left(t\right)$ and ${p}_{2}\left(t\right)$ are continuous and differentiable whenever this is required. A real function $f\left(t\right)$ defined in $\left(a,b\right)$ is said to be $\text{sub}\phantom{\rule{0.166667em}{0ex}}-\left(L\right)$ in $\left(a,b\right)$ if $f\left(t\right)\le S\left(f,{t}_{1},{t}_{2}\right)\left(t\right)$ for every $t$, ${t}_{1}$, ${t}_{2}$ such that $a<{t}_{1}, $S\left(f,{t}_{1},{t}_{2}\right)$ being the solution of $L\left(y\right)=0$ taking the values $f\left({t}_{1}\right)$ and $f\left({t}_{2}\right)$ at ${t}_{1}$ and ${t}_{2}$.

Further, let $I$ be a closed real interval, $k\in {ℕ}_{0}$ and ${L}_{n}:{C}^{k}\left(I\right)\to {C}^{k}\left(I\right)$ satisfying the following shape preserving property and asymptotic formula: (H1) for all $n\in ℕ$, ${L}_{n}$ is $k$-convex, (H2) there exist a sequence ${\lambda }_{n}$ of real positive numbers, and a function $p\in {C}^{k}\left(I\right)$ strictly positive on $\text{Int}\phantom{\rule{0.166667em}{0ex}}\left(I\right)$ such that for all $g\in {C}_{B}^{k}\left(I\right)$, $k+2$-times differentiable in some neighborhood of a point $x\in \text{Int}\phantom{\rule{0.166667em}{0ex}}\left(I\right)$, ${lim}_{n\to \infty }{\lambda }_{n}\left({D}^{k}{L}_{n}g\left(x\right)-{D}^{k}g\left(x\right)\right)={D}^{k}\left(p{D}^{2}g\right)\left(x\right)$.

The main result is:

Theorem. Let ${L}_{n}$ and $L$ be the operators which satisfy conditions H1 and H2, let $M\ge 0$ and let $a,b\in \text{Int}\phantom{\rule{0.166667em}{0ex}}\left(I\right)$ with $a. Then for $f,w\in {C}_{B}^{k}\left(I\right)$ (i) $M{D}^{k}w+{D}^{k}f$ is $\text{sub}-\left(L\right)$ in $\left(a,b\right)$ if and only if

${D}^{k}{L}_{n}f\left(x\right)-{D}^{k}f\left(x\right)\ge -M\left({D}^{k}{L}_{n}w\left(x\right)-{D}^{k}w\left(x\right)\right)+\sigma \left({\lambda }_{n}^{-1}\right),\phantom{\rule{4pt}{0ex}}x\in \left(a,b\right)·$

(ii) $M{D}^{k}w-{D}^{k}f$ is $\text{sub}-\left(L\right)$ in $\left(a,b\right)$ if and only if

${D}^{k}{L}_{n}f\left(x\right)-{D}^{k}f\left(x\right)\le M\left({D}^{k}{L}_{n}w\left(x\right)-{D}^{k}w\left(x\right)\right)+\sigma \left({\lambda }_{n}^{-1}\right),\phantom{\rule{4pt}{0ex}}x\in \left(a,b\right)·$

Finally, the theorem is applied to the well known approximation operators of Bernstein, Szász-Mirakyan, Mayer-König and Zeller, and Bleimann, Butzer and Hahn.

##### MSC:
 41A40 Saturation (approximations and expansions)