  It is well-known that sparse grid algorithm has been widely accepted as an
efficient tool to overcome the "curse of dimensionality" in some degree. In
this note, we first give the error estimate of hyperbolic cross (HC)
approximations with generalized Hermite functions. The exponential convergence
in both regular and optimized hyperbolic cross approximations has been shown.
Moreover, the error estimate of Hermite spectral method to high-dimensional
linear parabolic PDEs with HC approximations has been investigated in the
properly weighted Korobov spaces. The numerical result verifies the exponential
convergence of this approach.
