We revisit the Fourier analysis on the Heisenberg group ℍ d . Starting from the so-called Schrödinger representation and taking advantage of the projection with respect to the Hermite functions, we look at the Fourier transform of an integrable function f , as a function f ^ ℍ on the set ℍ ˜ d = d e f ℕ d × ℕ d × ℝ ∖ { 0 } . After observing that f ^ ℍ is uniformly continuous on ℍ ˜ d equipped with an appropriate distance d ^ , we extend the definition of f ^ ℍ to the completion ℍ ^ d of ℍ ˜ d . This new point of view provides a simple and explicit description of the Fourier transform of integrable functions, when the “vertical” frequency parameter tends to 0 . As an application, we prepare the ground for computing the Fourier transform of functions on ℍ d that are independent of the vertical variable.