Local saturation of conservative operators
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 \(I\subset \mathbb R\), \(i\in \mathbb N_0=\mathbb N\cup \{0\}\). 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 \(D^i f\geq 0\) 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 \(f(t)\leq S(f,t_1,t_2)(t)\) 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, \(k\in \mathbb N_0\) and \(L_n:C^k(I)\to C^k(I)\) satisfying the following shape preserving property and asymptotic formula: (H1) for all \(n\in \mathbb N\), \(L_n\) 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 \(M\geq 0\) 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 \[ D^kL_n f(x)-D^kf(x)\geq -M(D^kL_nw(x)-D^kw(x))+\sigma (\lambda^{-1}_n),\;x\in(a,b). \] (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.
scientific article; zbMATH DE number 2098983