H. Woznakowski
abstract:
We study tractability in the worst case setting of tensor product
linear operators defined over weighted tensor product Hilbert spaces.
Tractability means that the minimal number of evaluations needed to
reduce the initial error by a factor of $\e$ in the $d$-dimensional case
has a polynomial bound in both $\e^{-1}$ and $d$. By one evaluation we
mean the computation of an arbitrary continuous linear functional, and
the initial error is the norm of the linear operator~$S_d$ specifying the
$d$-dimensional problem.
We prove that nontrivial problems are tractable iff the dimension of the
image under~$S_1$ (the one-dimensional version of~$S_d$) of the
unweighted part of the Hilbert space is one, and the weights
of the Hilbert spaces, as well as the singular values of the linear
operator $S_1$, go to zero polynomially fast with their indices.