Approximation of Functions of Few Variables in High Dimensions

DeVore, Ronald; Petrova, Guergana; Wojtaszczyk, Przemyslaw
February 2011
Constructive Approximation;Feb2011, Vol. 33 Issue 1, p125
Academic Journal
Let f be a continuous function defined on O:=[0,1] which depends on only l coordinate variables, $f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}})$. We assume that we are given m and are allowed to ask for the values of f at m points in O. If g is in Lip1 and the coordinates i,..., i are known to us, then by asking for the values of f at m= L uniformly spaced points, we could recover f to the accuracy | g| L in the norm of C( O). This paper studies whether we can obtain similar results when the coordinates i,..., i are not known to us. A prototypical result of this paper is that by asking for C( l) L(log N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy | g| L when g?Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance e (which is not known) by functions of l variables.


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