  Faraday rotation measure (RM) synthesis is an important tool to study and
analyze galactic and extra-galactic magnetic fields. Since there is a Fourier
relation between the Faraday dispersion function and the polarized radio
emission, full reconstruction of the dispersion function requires knowledge of
the polarized radio emission at both positive and negative square wavelengths
$\lambda^2$. However, one can only make observations for $\lambda^2 > 0$.
Furthermore observations are possible only for a limited range of wavelengths.
Thus reconstructing the Faraday dispersion function from these limited
measurements is ill-conditioned. In this paper, we propose three new
reconstruction algorithms for RM synthesis based upon compressive
sensing/sampling (CS). These algorithms are designed to be appropriate for
Faraday thin sources only, thick sources only, and mixed sources respectively.
Both visual and numerical results show that the new RM synthesis methods
provide superior reconstructions of both magnitude and phase information than
RM-CLEAN
