= Ill-posedness of numerical differentiation
{title2=$\|f_k\|_\infty\to0\text{ need not imply }\|f_k'\|_\infty\to0$}
Recovering a <derivative> from values controlled only in the <uniform norm> is an <ill-posed problem>: the differentiation map is not <continuous> in those norms. On $[0,1]$, take $f_k(x)=\sin(kx)/k$. Then $\|f_k\|_\infty\leq1/k\to0$, while $f_k'(x)=\cos(kx)$ has <uniform norm> one for every $k$. Arbitrarily small value perturbations can therefore conceal nonvanishing <derivative> changes.
<Numerical differentiation> can still be useful when additional smoothness bounds, a suitable sampling scale, or a <regularization> method control the admissible changes. The basic <noise amplification by differentiation of a linear interpolant> gives a finite-data version: endpoint uncertainty of size $\delta$ produces <derivative> uncertainty of order $\delta/h$ at spacing $h$.
Back to article page
