16.3 Experimental data space

The experimental data  $\Psi_{\! e}(\mathbf{u})$ are blurred values of  $\widehat{\Phi}_o(\mathbf{u})$ on a finite list of frequencies in the Fourier domain:

$$\mathcal{L}_e = \{ \mathbf{u}_e(1), \mathbf{u}_e(2), \dots, \mathbf{u}_e(N_e)\}.$$ (16.6)

As the object function of interest $\Phi_{\! o}$ is a real-valued function, it is natural to define  $\Psi_{\! e}(-\mathbf{u})$ as the complex conjugate of  $\Psi_{\! e}(\mathbf{u})$. The experimental frequency list  $\mathcal{L}_e$ is defined consequently: if  $\mathbf{u}\in\mathcal{L}_e$, then  $-\mathbf{u}\in\mathcal{L}_e$ (except for the null frequency  $\mathbf{u}=\mathbf{0}$: in the convention adopted here, either it does not lie in  $\mathcal{L}_e$, or there exists only one occurrence of this point). The experimental frequency coverage generated by  $\mathcal{L}_e$ is therefore centrosymmetric (see Fig. 16.3).

Figure 16.3: An example of an experimental frequency coverage provided by the IRAM interferometer. Here, the number of points $N_e$ in the experimental frequency list  $\mathcal{L}_e$ is equal to $2862$.

The experimental data vector  $\Psi_{\! e}$ lies in the experimental data space $K_e$, the real Euclidian space underlying the space of complex-valued functions $\psi$ on  $\mathcal{L}_e$, such that  $\psi(-\mathbf{u}) = \bar{\psi}(\mathbf{u})$. The dimension of this space is equal to $N_e$: the number of points in the experimental frequency list  $\mathcal{L}_e$.