Abstract
Let f be a continuous function defined on Ω:=[0,1]N 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 Ω. If g is in Lip1 and the coordinates i 1,…,i ℓ are known to us, then by asking for the values of f at m=L ℓ uniformly spaced points, we could recover f to the accuracy |g|Lip1 L −1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i 1,…,i ℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ℓ(log 2 N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1 L −1 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 ε (which is not known) by functions of ℓ variables.
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Communicated by Albert Cohen.
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contracts W911NF-05-1-0227 and W911NF-07-1-0185; the NSF Grants DMS-0810869 and DMS 0915231; and the Polish MNiSW grant N201 269335.
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DeVore, R., Petrova, G. & Wojtaszczyk, P. Approximation of Functions of Few Variables in High Dimensions. Constr Approx 33, 125–143 (2011). https://doi.org/10.1007/s00365-010-9105-8
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DOI: https://doi.org/10.1007/s00365-010-9105-8