Three conjectures on Shepard interpolatory operators
For \(f\in C[0,1] \), the Shepard interpolatory operator \(S_{n,\lambda} \) has the images \[ (S_{n,\lambda}f)(x)=\frac{ \sum_{k=0}^{n}| x-k/n| ^{- \lambda}f(k/n) } {\sum_{k=0}^{n} | x-k/n| ^{- \lambda}}, \] where \(n=1,2,\dots; \lambda \geq 1\). The main results are the following: Theorem 1: There is at least a function \( f , f\in C^1[0,1] \) such that \[ \| f-S_{n,2}f\| =O(1/n), \] while \(\int_{0}^{1}{ t^{-1}\omega(f',t)dt} = \infty . \) Theorem 2: For \(1< \lambda < 2 \), \[ \| f-S_{n,\lambda }f\| = o (n^{1 - \lambda}) \text{ implies } f=\text{const.} \] Moreover, there is a function \(f\in C[0,1] \) such that \[ \| f-S_{n,\lambda}f\| =O(n^{1- \lambda}), \] while \(\int_{0}^{1}t^{-1}\omega(f,t)dt = \infty\). Theorem 3: If \(\| f-S_{n,1} f\|= o (\log^{-1}{n})\), then \(f=\text{const.}\) Also, there is a function \(f\in C[0,1] \) such that \[ \| f-S_{n,1}f\| =O( \log^{-1}{n}), \] while \( \int_{0}^{1}t^{-1}\omega (f,t)dt = \infty\).
scientific article; zbMATH DE number 1197061
scientific article