THE DISCRETE K-FUNCTIONAL AND SPLINE SMOOTHING OF NOISY DATA

Estimation of a function f from a finite sample y equals left bracket f(x//i) plus epsilon //i right bracket , x//i an element of left bracket a, b right bracket , subject to random noise epsilon //i, is a basic problem of numerical approximation theory. This paper defines a discrete analog, k//m(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.