The multiple Hermite series in Rn are investigated by the Riesz summability
method of order α>(n−1)/2. More precisely, localization theorems for some classes of functions
are proved and sharp sufficient conditions are given. Thus the classical Szegö results are extended
to the n-dimensional case. In particular, for these classes of functions the localization principle
and summability on the Lebesgue set are established.