Linear combinations of Lagrange's interpolation polynomial (Q2703596)

Because the Lagrange interpolation polynomial does not converge uniformly for an arbitrary continuous function, in this paper, the interpolated functions are linearly combined in Lagrange interpolation polynomials. An operator \(A_{n,r}(f;x)\) is constructed based on the zeros of \((1-x)W_n(x)\) as the interpolation nodes, where \(W_n(x)=\sin\frac{(2n+1)\theta}{2}/\sin\frac{\theta}{2},x=\cos \theta, \theta\in [0,\pi]\). It converges uniformly to the concerned arbitrary continuous,and derivable function \(f(x)\in C^l_{[-1,1]}\) \((0\leq l\leq r)\) and the convergence order is \(|A_{n,r}(f;x)-f(x)|=O[E_n(f)+\frac{1}{n^l}\omega(f^{(l)},\frac 1n)+\frac{1}{n^{l+1}}]\), and the convergence order is best (where \(r\) is an odd natural number).