  Decomposition systems with rapidly decaying elements (needlets) based on
Hermite functions are introduced and explored. It is proved that the
Triebel-Lizorkin and Besov spaces on $\R^d$ induced by Hermite expansions can
be characterized in terms of the needlet coefficients. It is also shown that
the Hermite Triebel-Lizorkin and Besov spaces are, in general, different from
the respective classical spaces.
