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In this paper we consider integration and $L_2$-approximation for functionsover $\RR^s$ from weighted Hermite spaces. The first part of the paper isdevoted to a comparison of several weighted Hermite spaces that appear inliterature, which is interesting on its own. Then we study tractability of theintegration and $L_2$-approximation problem for the introduced Hermite spaces,which describes the growth rate of the information complexity when the errorthreshold $\varepsilon$ tends to 0 and the problem dimension $s$ grows toinfinity. Our main results are characterizations of tractability in terms ofthe involved weights, which model the importance of the successive coordinatedirections for functions from the weighted Hermite spaces.

Tractability of $L_2$-approximation and integration in weighted Hermite spaces of finite smoothness