Almost Optimal Differentiation Using Noisy Data

✍ Scribed by Klaus Ritter

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 561 KB
• Volume: 86
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.0071

✦ Synopsis

We study differentiation of functions f based on noisy data f (t i )+= i . We recover f (k) either at a single point or on the interval [0, 1] in L 2 -norm. Under stochastic assumptions on f and = i , we determine the order of the errors of the best linear methods which use n noisy function values. Polynomial interpolation for the pointwise problem and smoothing splines for the problem in L 2 -norm are shown to be almost optimal. The analysis involves worst case estimates in reproducing kernel Hilbert spaces and a Landau inequality.

1996 Academic Press, Inc.

where g i # R for recovering f (k) (t), and g i # L 2 ([0, 1]) for recovering f (k) .

Assumptions on the function f and on the noise = are needed to derive error bounds for methods S n . In this paper = 1 , ..., = n are uncorrelated random variables with zero mean and common variance _ 2 >0, i.e., E(= i )=0 article no.