@article {IOPORT.05864162,
    author       = {K\"ammerer, Lutz and Kunis, Stefan},
    title        = {On the stability of the hyperbolic cross discrete Fourier transform.},
    year         = {2011},
    journal      = {Numerische Mathematik},
    volume       = {117},
    number       = {3},
    issn         = {0029-599X},
    pages        = {581-600},
    publisher    = {Springer-Verlag, Berlin},
    doi          = {10.1007/s00211-010-0322-7},
    abstract     = {A straightforward discretisation of problems in $d$ spatial dimensions with $2^n$ grid points in each coordinate leads to an exponential growth $2^{dn}$ in the number of degrees of freedom. For moderately high dimensional problems, sparse grid approximations allow for a severe decrease in the number of used Fourier coefficients to represent 1-periodic functions with bounded mixed derivatives. In this paper, the authors estimate the condition number of the modified Fourier matrix $$ F_n^d = (\exp(2\pi i \, k\cdot x))_{ x\in S_n^d,\, k \in H_n^d} $$ for increasing refinement $n\in \mathbb N$ and $d\ge 2$. Here $S_n^d$ denotes the sparse grid of $[0,1)^d$ and $H_n^d$ is the hyperbolic cross in the frequency domain $ {\mathbb Z}^d$. Using a Boolean sum decomposition of $( F_n^d )^{-1}$, lower and upper estimates of $\| F_n^d \|_2$ and $\|(F_n^d )^{-1}\|_2$ for fixed dimension $d$ and variable $n \ge d$ are presented such that the condition number of $F_n^d $ scales approximately like $|H_n^d|^{1/2}$. For fixed refinement $n$ and variable $d\ge n$, the spectral norms of $ F_n^d $ and $(F_n^d )^{-1}$ are estimated too. In this case, the condition number of $F_n^d $ scales approximately like $|H_n^d|^2$. These results are refined for $d=2$ and $n=1$, respectively. All results are illustrated by numerical experiments.},
    reviewer     = {Manfred Tasche (Rostock)},
    identifier   = {05864162},
}