Convergence Rates of Approximation by Translates
Abstract
In this paper, the authors consider the problem of approximating a function belonging to some function space PHI by a linear combination of n translates of a given function G. Using a lemma by Jones (1990) and Barron (1991), they show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the error is O(1 over the square root of n) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev type, in which the number of weak derivatives is required to be larger than the number of dimensions. They give results both for approximation in the L(sub 2) norm and in the L(sub infinity) norm. The interesting feature of these results is that, thanks to the constructive nature of Jones' and Barron's lemma, an iterative procedure is defined that can achieve this rate.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1992
- Accession Number
- ADA260100
Entities
People
- Federico Girosi
- Gabriele Anzellotti
Organizations
- Massachusetts Institute of Technology