The aim of this presentation is to give some results concerning the output tracking of the one dimensional heat equation, subject to a control applied on the boundary. The tracking problem can be informally defined as: characterize what are the functions $t \mapsto z(t,0)$, where $z$ ranges on all the controlled solutions of the equation. Invoking the literature on the control of the heat equation it is possible to give a partial answer in terms of Gevrey spaces, but such classes are not enough to fully answer the question. After passing the equation in the frequency domain, we will translate the tracking problem in the realm of Hardy spaces. Therein, everything boils down to the following question: does the spectrum (i.e. the support of the Fourier transform) of a function decreases if we multiply it by a power series? In full generality this question has a negative answer, but we conjecture that under additional hypotheses this is true. This is joint work with Pierre Lissy (CERMICS), Olivier Glass (CEREMADE) and Swann Marx (LS2N).