Function approximation and integration on the wiener space with noisy data

Abstract We study the approximation and integration problems for r times continuously differentiable scalar functions, based on noisy observations of the values of the function or its derivatives at n points. The noise corresponding to each observation has normal distribution with variance σ2. We consider the average error with respect to the noise and r-fold Wiener measure on the function space. We show that for r = 0 the nth minimal error is asymptotically equal to 1 √6n + p( σ 2 (4n) ) 1 4 where 1 √3 ≤ p, q ≤ 1 . For both problems the optimal sample points are equidistant. For r ≥ 1 the corresponding nth minimal errors are proportional to n −(r+ 1 2 ) + σ √n and n −(r+1) + σ √n .