A new methodology for Fourier synthesis: Fourier Interpolation and Reconstruction via Shannon-type Techniques (FIRST).

We present a new approach to the problems of Fourier synthesis in the experimental context of aperture synthesis. Depending on what is emphasized, this method is called FIRST or WIPE:FIRST for the principles (Fourier Interpolation and Reconstruction via Shannon-type Techniques), and WIPE for the corresponding deconvolution method (WIPE reminds of CLEAN, a well-known deconvolution technique in astronomy). The regularization principle of FIRST refers to the Shannon sampling formula and to theoretical considerations related to multiresolution analysis. To describe the imaging kernel of FIRST (WIPE), the authors adopt a terminology derived from that of CLEAN. At each iteration of the selected constructive process (conjugate gradients for example), WIPE compares the dusty map with the dusty map of the model. In the corresponding truncated discrete convolution, the discrete point-spread function, the dusty beam, has two components: the traditional dirty beam and the regularization beam. Besides the clarity of the principles, the authors show, in a concrete manner, the advantages of WIPE over CLEAN. In a more general way, they also indicate how FIRST can be used for analysing the results provided by other methods