Tractability of multivariate approximation over a weighted unanchored Sobolev space: Smoothness sometimes hurts

We study d-variate L2-approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m , even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unlessm = 1, i.e., unless the smoothness is minimal. We then show that our problem can be tractable ifm = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε−1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly This research was supported in part by a Fordham University Faculty Fellowship. †This research was supported in part by the National Science Foundation.