Convergence Rates of Approximation by Translates

Abstract

We consider the problem of approximating a function belonging to some function space by a linear combination of $n$ translates of a given function. Using a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces for which the rate of convergence to zero of the error is $O({1 \over \sqrt n})$ 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.