We describe an interpolation/extrapolation procedure that reconstructs X-ray maps of solar flares using as input data sparse samples of the Fourier transform of the radiation flux, named visibilities. The algorithm is based on two steps: in the first step the performance of an interpolation routine is optimized by representing the visibilities according to favorable coordinates in the frequency plane. In the second step two extrapolation schemes are introduced, respectively based on the projection and the thresholding of the Landweber iterative method. The procedure is validated against realistic synthetic visibilities and applied to experimental measurements provided by the NASA satellite Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).