Approximation of linear functionals on a banach space with a Gaussian measure

AbstractWe study approximation of linear functionals on separable Banach spaces equipped with a Gaussian measure. We study optimal information and optimal algorithms in average case, probabilistic, and asymptotic settings, for a general error criterion. We prove that adaptive information is not more powerful than nonadaptive information and that μ-spline algorithms, which are linear, are optimal in all three settings. Some of these results hold for approximation of linear operators. We specialize our results to the space of functions with continuous rth derivatives, equipped with a Wiener measure. In particular, we show that the natural splines of degree 2r + I yield the optimal algorithms. We apply the general results to the problem of integration