Symmetric node pairing bound for interpolation on the unit interval (${\sup }_{0\le x\le 1}\left|{\prod }_j(x-x_j)\right|\le 2^{-m}$)

Suppose $m$ interpolation nodes in $[0,1]$ occur in pairs $a,1-a$, with the middle node $1/2$ if $m$ is odd. For each pair,because both squared quantities lie between zero and one quarter. An unpaired middle factor has magnitude at most one half. Multiplication gives the displayed bound on the nodal polynomial.

The polynomial interpolation remainder therefore satisfies $\Vert f-p_{m-1}{\Vert }_{\infty }\le \Vert f^{(m)}{\Vert }_{\infty }/[2^{m}m!]$ for $f\in C^m[0,1]$. For $f(x)=\sin 2x$, the derivative bound $\Vert f^{(m)}{\Vert }_{\infty }\le 2^m$ gives the particularly simple error bound $1/m!$. This proves uniform convergence for these symmetric-node interpolants of that function. It is not a general convergence theorem for arbitrary continuous interpolation data.