On (0,1,2) Trigonometric Interpolation

Abstract

The aim of this paper is to discuss the case of $(0,1,2)$ interpolation by trigonometric polynomial on the zeros of $\sin mx$ at the point $x_k$=$\frac{2\pi k}{n}$, where $k=0,1,2,\dots,n-1$, where n is even $(n=2m)$ and the convergence behavior of this trigonometric polynomial. Let $R_n(x)$ be a trigonometric polynomial of order such that \begin{align*} R_n(x_k)&=a_k\\ R'_n(x_k)&=b_k\\ R''_n(x_k)&=c_k \end{align*} where $x_k$=$\frac{2\pi k}{n}$, $k=0,1,2,\dots,n-1$ and $n$ is even $(n=2m)$.