  In this paper we describe a general method to generate superoscillatory
functions of several variables starting from a superoscillating sequence of one
variable. Our results are based on the study of suitable infinite order
differential operators on holomorphic functions with growth conditions of
exponential type, where additional constraints are required when dealing with
infinite order differential operators whose symbol is a function that is
holomorphic in some open set, but not necessarily entire. The results proved
for the superoscillating sequence in several variables are extended to
sequences of supershifts in several variables.
