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Starting with far field data of time-harmonic acoustic or electromagneticwaves radiated by a collection of compactly supported sources intwo-dimensional free space, we develop criteria and algorithms for the recoveryof the far field components radiated by each of the individual sources, and thesimultaneous restoration of missing data segments. Although both parts of thisinverse problem are severely ill-conditioned in general, we give preciseconditions relating the wavelength, the diameters of the supports of theindividual source components and the distances between them, and the size ofthe missing data segments, which guarantee that stable recovery in presence ofnoise is possible. The only additional requirement is that a priori informationon the approximate location of the individual sources is available. We giveanalytic and numerical examples to confirm the sharpness of our results and toillustrate the performance of corresponding reconstruction algorithms, and wediscuss consequences for stability and resolution in inverse source and inversescattering problems.

Uncertainty principles for inverse source problems, far field splitting and data completion