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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 $I\subset \Bbb R$, $i\in \Bbb N_0=\Bbb N\cup \{0\}$. Let $C^i(I)$ 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 $C(I)=C^0(I)$ the space of all continuous functions defined on $I$ and $D^0$ the identity operator, respectively. Besides, $C^i_B(I)$ denotes the subspace formed by the functions of $C^i(I)$ which are bounded on $I$. A function $f\in C^i(I)$ 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 $L(y)=0$ denote the second-order linear differential equation $L(y)\equiv D^2 y+p_1(t)D^1 y+p_2(t)y=0$. Consider an interval $(a,b)$ such that $L(y)=0$ has a unique solution, continuous in $(a,b)$, taking any given real values $y_1$ and $y_2$ at any two given $t_1$ and $t_2$ within $(a,b)$. Assume that $p_1(t)$ and $p_2(t)$ are continuous and differentiable whenever this is required. A real function $f(t)$ defined in $(a,b)$ is said to be $\text{sub}\,-(L)$ in $(a,b)$ if $f(t)\le S(f,t_1,t_2)(t)$ for every $t$, $t_1$, $t_2$ such that $a<t_1<t<t_2<b$, $S(f,t_1,t_2)$ being the solution of $L(y)=0$ taking the values $f(t_1)$ and $f(t_2)$ at $t_1$ and $t_2$. Further, let $I$ be a closed real interval, $k\in \Bbb N_0$ and $L_n:C^k(I)\to C^k(I)$ satisfying the following shape preserving property and asymptotic formula: (H1) for all $n\in \Bbb N$, $L_n$ is $k$-convex, (H2) there exist a sequence $\lambda_n$ of real positive numbers, and a function $p\in C^k(I)$ strictly positive on $\text{Int}\,(I)$ such that for all $g\in C^k_B(I)$, $k+2$-times differentiable in some neighborhood of a point $x\in \text{Int}\,(I)$, $\lim_{n\to\infty}\lambda_n(D^k L_ng(x)-D^kg(x))=D^k(pD^2g)(x)$. 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 $a,b\in \text{Int}\, (I)$ with $a<b$. Then for $f,w\in C^k_B(I)$ (i) $MD^kw+D^k f$ is $\text{sub}-(L)$ in $(a,b)$ if and only if $$ D^kL_n f(x)-D^kf(x)\ge -M(D^kL_nw(x)-D^kw(x))+\sigma (\lambda^{-1}_n),\ x\in(a,b). $$ (ii) $MD^kw-D^kf$ is $\text{sub}-(L)$ in $(a,b)$ if and only if $$ D^kL_nf(x)-D^kf(x)\leq M(D^kL_n w(x)-D^kw(x)) +\sigma(\lambda^{-1}_n), \ x\in(a,b). $$ Finally, the theorem is applied to the well known approximation operators of Bernstein, Szász-Mirakyan, Mayer-König and Zeller, and Bleimann, Butzer and Hahn.

41A40Saturation (approximations and expansions)
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