  This paper is the first part of a two-paper series whose aim is to give a
thorough account on Connes' pseudodifferential calculus on noncommutative tori.
This pseudodifferential calculus has been used in numerous recent papers, but a
detailed description is still missing. In this paper, we focus on constructing
an oscillating integral for noncommutative tori and laying down the main
functional analysis ground for understanding Connes' pseudodifferential
calculus. In particular, this allows us to give a precise explanation of the
definition of pseudodifferential operators on noncommutative tori. More
generally, this paper introduces the main technical tools that are used in the
2nd part of the series to derive the main properties of these operators. In
addition, we establish the equivalence between our class of operators and the
toroidal pseudo differential operators considered by other authors.
