= Symmetric node pairing bound for interpolation on the unit interval
{title2=$\sup_{0\le x\le1}\left|\prod_j(x-x_j)\right|\le2^{-m}$}
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,
$$
|(x-a)(x-(1-a))|=|(x-1/2)^2-(a-1/2)^2|\le1/4,
$$
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 $\|f-p_{m-1}\|_\infty\le\|f^{(m)}\|_\infty/[2^m m!]$ for $f\in C^m[0,1]$. For $f(x)=\sin2x$, the <derivative> bound $\|f^{(m)}\|_\infty\le2^m$ 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.
Back to article page
