On certain saturation problems
In the present paper, the author studies certain saturation problems on the interpolatory linear operators \[ (L_nf)(x)=\sum_{0\leq k\leq n}f(k/n)|x-k/n|^{-r}/\sum_{0\leq k\leq n} |x-k/n/|^{-r}; \quad 0\leq x\leq 1,\quad n\geq 1, \] where \(r>2\) is a fixed real number, \(f\in C[0,1]\), and gives two theorems on it. His main result is as follows: ``Theorem: Let \(0<x_0<1\) be a fixed irrational number, \(\{y_r\}^{\infty}_{r-1}\) be an arbitrary sequence with \(y_r\neq x_0\), \(r=1,2,...\), \(\lim_{r\to \infty}y_r=x_0\). Further let \(0<p*\leq 1/3\) (real), \(p,q>0\), \((p,q)=1\) (integers), \(0\leq \gamma <p\), \(0\leq \delta <q\) (reals) be fixed numbers. Then there exist a sequence \(\{x_k\}\subset \{y_r\}\) and positive integers \(\{\ell_k\}^{\infty}_{k=1}\) and \(\{n_k\}^{\infty}_{k=1}\) with \(1<n_1<n_2<...\). i.e. \(\lim_{k\to \infty}n_k=\infty\) such that relations \[ |x_0-(p\ell_k+\gamma)/(qn_k+\delta)| = o(1/n_k), \quad k=1,2,... , \] and \[ p*/2n_k \leq |x_k-(p\ell_k+\gamma)/(qn_k+\delta)| \leq (2p+2)p/n_k, \quad k=1,2,..., \] hold true.
scientific article; zbMATH DE number 4117150
scientific article