  In this paper, we present a rigorous analysis of root-exponential convergence
of Hermite approximations, including projection and interpolation methods, for
functions that are analytic in an infinite strip containing the real axis and
satisfy certain restrictions on the asymptotic behavior at infinity within this
strip. Asymptotically sharp error bounds in the weighted and maximum norms are
derived. The key ingredients of our analysis are some remarkable contour
integral representations for the Hermite coefficients and the remainder of
Hermite spectral interpolations. Further extensions to Gauss--Hermite
quadrature, Hermite spectral differentiations, generalized Hermite spectral
approximations and the scaling factor of Hermite approximation are also
discussed. Numerical experiments confirm our theoretical results.
