Rates for approximation of unbounded functions by positive linear operators
This paper is concerned with an extension of the Bohman-Korovkin theorem to the approximation of unbounded functions. In particular, the following situation is considered: Let \(I=(a,b)\) where -\(\infty \leq a\leq b\leq \infty\) and \(I_ 1=[a_ 1,b_ 1]\subseteq I\). Let \(g_ a(t)\), \(g_ b(t)\) be given functions which are positive, strictly convex on I, continuously differentiable on \(I_ 1\), and satisfy \(g_ b(t)=O(g_ a(t))\) as \(t\to a+0\) and \(g_ a(t)=O(g_ b(t))\) as \(t\to B-0\). If \(a=- \infty\) and \(b=\infty\), respectively, it is assumed that \(\lim_{t\to - \infty}g_ a(t)_{/| t|}=\infty\) and \(\lim_{t\to \infty}g_ b(t)_{/| t|}=\infty,\) respectively. As Theorem 2.1 the following is proved: Let \(\{L_ n\}\) be a sequence of positive linear operators on \(C(I_ 1,g_ a,g_ b)\) which is the space of functions f on I with the following properties: 1) f is bounded on every compact subset of I, 2) f is continuous on \(I_ 1\), 3) \(f(t)=O(g_ a(t))\) as \(t\to a+0\) and \(f(t)=O(g_ b(t))\) as \(t\to b-0\). The operators \(L_ n\) are required to satisfy \(L_ n(g_ a,x)<\infty\) and \(L_ n(g_ b,x)<\infty\) for all \(x\in I_ 1\) and every \(n\in {\mathbb{N}}\). Assertion: If \(L_ n(t^ i,x)\to x^ i\) (for \(i=0,1)\) and \(L_ n(g_ s,x)\to g_ s(x)\) (for \(s=a,b)\) for some \(x\in I_ 1\) and uniformly for all \(x\in I_ 1\), respectively, then for every \(f\in C(I_ 1,g_ a,g_ b)\) the sequence \(\{L_ n(f,x)\}\) convergences to f(x) at \(x\in I_ 1\) and uniformly for all \(x\in I_ 1\), respectively. In Theorem 2.2 error estimates are presented from which convergence rates can be derived. The results are demonstrated by six examples which include modifications of some well-known positive linear operators.
scientific article; zbMATH DE number 4136683
scientific article