Approximation with exponential weights in \([-1,1]\)
Let \(w(x):=\exp(-1/(1-x^2)^\alpha)\), \(\alpha>0\), \(|x|<1\), and denote by \(C_w\) the space of continuous functions on \((-1,1)\) for which \(\|wf\|:=\sup_{|x|<1}w(x)|f(x|\) is finite. Fix \(0\leq\beta<1\) and let \(x_{k,n}:=x_k:=(1-((2k)/n)^{1/(1-\beta)})(1-n^{-1/(1+\alpha-\beta)})\), \(k=0,\dots,[n/2]\), and \(x_{k,n}:=-x_{n-k}\), \(k=[n/2]+1,\dots,n\). The Shepard-type operator \(S_n\) is defined by \[ S_n(f;x):=\frac{\sum_{k=0}^n\exp(|x-x_k|^{-s})f(x_k)}{\sum_{k=0}^n\exp(|x-x_k|^{-s})}, \] \(s>0\). It is a positive operator, interpolating \(f\) at the \(x_k\)'s and preserving constants. Also \(S'_n(f;x_k)=0\), \(0\leq k\leq n\). The authors define a weighted modulus of continuity \(\omega^\varphi(f;t)_w\), where \(\varphi(x):=(1-x^2)^\beta\), and an equivalent K-functional \(K^\varphi(f;t)_w:=\inf_{g\in AC}\{\|w(f-g)\|+t\|w\varphi g\|\}\), and show that the weighted approximation of \(f\) by \(S_n(f)\) can be estimated from above by \(C\omega^\varphi(f;1/n)_w\).
scientific article; zbMATH DE number 1836773