Almost optimal differentiation using noisy data
Differentiation of functions \(f\) is studied if only inaccurate data \(f(t_i)+\varepsilon_i\) are available, the \(f\) being stochastic and corrupted by some noise \(\varepsilon_i\), the function \(f\) being observed at a finite number of points. Expressions are derived for the derivative \(f^{(k)}\) either at a single point \(t\in [0,1]\) or on the whole interval \([0,1]\). In this latter case the distance from \(f^{(k)}\) in \(L_2\)-norm is considered. Optimality of smoothing spline methods is discussed. An expression is also found for the order of minimal errors for recovering \(f^{(k)}\) in the interval \([0,1]\).
scientific article; zbMATH DE number 938744