Combined Shepard operators with Chebyshev nodes

Abstract

In this paper we study combined Shepard-Lagrange univariate interpolation operator\[S_{n,\mu}^{L,m}(Y;f,x):=S_{n,\mu}^{L,m}(f,x)=\frac{\sum\limits_{k=0}^{n+1}\left\vert x-y_{n,k}\right\vert ^{-\mu}(L_{m}f)(x,y_{n,k})}{\sum\limits_{k=0}^{n+1}\left\vert x-y_{n,k}\right\vert ^{-\mu}},\]where \((y_{n,k})\) are the interpolation nodes and \((L_{m}f)(x;y_{n,k})\) is the Lagrange interpolation polynomial with nodes \( y_{n,k},\, y_{n,k+1}, \, y_{n,k+2}, \ldots, \, y_{n,k+m} \), when the interpolation nodes \((y_{n,k})_{k=\overline{1,n}}\) are the zeros of first kind Chebyshev polynomial completed with \(y_{n,0}=-1\)and \(y_{n,n+1}=1\). We give a direct proof for error estimation and some numerical examples.