  It has been found empirically that quasi-Monte Carlo methods are often
efficient for very high-dimensional problems, that is, with dimension in the
hundreds or even thousands. The common explanation for this surprising fact is
that those functions for which this holds true behave rather like
low-dimensional functions in that only few of the coordinates have a size- able
influence on its value. However, this statement may be true only after applying
a suitable orthogonal transform to the input data, like utilizing the Brownian
bridge construction or principal component analysis construction. We study the
effect of general orthogonal transforms on functions on $\mathbb{R}^d$ which
are ele- ments of certain weighted reproducing kernel Hilbert spaces. The
notion of Hermite spaces is defined and it is shown that there are examples
which admit tractability of integration. We translate the action of the
orthogonal transform of $\mathbb{R}^d$ into an action on the Hermite
coefficients and we give examples where orthogonal transforms have a dramatic
effect on the weighted norm, thus providing an explanation for the efficiency
of using suitable orthogonal transforms.
