Converse theorems for multidimensional Kantorovich operators
The author devotes most of the paper to a detailed treatment of direct and inverse theorems for Kantorovich type operators \(K_ n\) defined on \(L_ p(S)\) where \(S\) is the triangle \(\{(x,y): x,y\geq 0,\;x+ y\leq 1\}\) by \[ K_ n(f,x,y)= \sum_{k+ m\leq n} {n\choose k}{n-k\choose m} x^ k y^ m(1- x- y)^{n-k-m} 2(n+1)^ 2\iint_{\Delta_{k,m}} f(s,t)ds dt, \] where \[ \Delta_{k,m}= \left\{(x,y): {k\over n+1}\leq x,\;{m\over n+1}\leq y,\;x+ y\leq {k+m+1\over n+1}\right\}. \] Theorem 1 states that for \(1\leq p< \infty\), \(0< \alpha< 1\) and \(f\in L_ p(S)\) the condition \[ \| K_ n f- f\|_ p= O(n^{-\alpha}) \] is equivalent to each of several smoothness conditions, one involving a Peetre \(K\)-functional, the others second order difference operators. Theorem 2 states that for \(f\in C(S)\), \(0< \alpha<1\), if \(B_ n\) is a multidimensional Bernstein operator then the two conditions \[ \| K_ n f- f\|_ \infty= O(n^{-\alpha})\quad\text{and}\quad \| B_ n f- f\|_ \infty= O(n^{-\alpha}) \] are equivalent and equivalent to each of several smoothness conditions. It is shown that Theorem 2 does not extend to the case \(\alpha= 1\). Finally \(m\)-dimensional Kantorovich operators are defined for an \(m\)-simplex and for an \(m\)-dimensional cube and results for these operators are stated without proof.
scientific article; zbMATH DE number 500213