In the 1970s, H.O Fattorini and D.L Russell, [4], [5], linked the method of moments to the study of the controllability at time T > 0 of the heat equation. The basic argument is the construction and estimation of bi-orthogonal families in L2(0T) a {exp- landakt, k>= 1} where landak, k>= 1 is the sequence of eigenvalues of Sturm Liouville operators. Muntzs theorem and Weyls formula require us to restrict ourselves to the dimension 1 of space. In recent years works such as, for example, [1], [3], [2] have shown that even in the case of parabolic equations, controllability to 0 at time T could require T >=T0 with T0 > 0. These results, based on high frequency phenomena, are obtained by deepening the work of [5] on the construction and estimation of families bi-orthogonal in L2(0T) a { exp- landakt,landa in F} where F contained in C. Consequently, they are restricted to the dimension 1 of space. The objective of this talk is to take a rst step in the construction of bi-orthogonal families in L2((0T)Xw) (in space-time) a  { exp- landakt,landa in F} . F in R, Phi landa,landa in F contained in L2(Omega ),w contained in Omega,Omega contained in  Rd,d >=1.