On the randomized complexity of Banach space valued integration
Volume 223 / 2014
Studia Mathematica 223 (2014), 205-215
MSC: Primary 65D30; Secondary 46B07, 46N40, 65C05, 65Y20.
DOI: 10.4064/sm223-3-2
Abstract
We study the complexity of Banach space valued integration in the randomized setting. We are concerned with $r$ times continuously differentiable functions on the $d$-dimensional unit cube $Q$, with values in a Banach space $X$, and investigate the relation of the optimal convergence rate to the geometry of $X$. It turns out that the $n$th minimal errors are bounded by $cn^{-r/d-1+1/p}$ if and only if $X$ is of equal norm type $p$.
