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We solve the inverse problems to recover Dirac systems on an interval orsemiaxis from their spectral functions (matrix valued functions) for the caseof locally square-integrable potentials. Direct problems in terms of spectralfunctions are treated as well. Moreover, we present necessary and sufficientconditions on the given distribution matrix valued function to be a spectralfunction of some Dirac system with a locally square-integrable potential.Interesting connections with Paley-Wiener sampling measures appear in the caseof scalar spectral functions.

Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions