The Discrete k-Functional and Spline Smoothing of Noisy Data

Estimation of a function f from a finite sample ${\bf y} = [f(x_i ) + \varepsilon _i ],x_i \in [a,b]$, subject to random noise $\varepsilon _i $, is a basic problem of numerical approximation theory. This paper defines a discrete analog, $k_m ({\bf y},\lambda )$, of Peetre’s K-functional, which relates to spline smoothing. We show how to use $k_m $ and its connection to the mth order modulus of continuity to assess the smoothness of f and to choose a good smoothing spline approximation to f and some of its derivatives.