  Green's functions of fermions are described by matrix-valued
Herglotz-Nevanlinna functions. Since analytic continuation is fundamentally an
ill-posed problem, the causal space described by the matrix-valued
Herglotz-Nevanlinna structure can be instrumental in improving the accuracy and
in enhancing the robustness with respect to noise. We demonstrate a
three-pronged procedure for robust analytic continuation called PES: (1)
Projection of data to the causal space. (2) Estimation of pole locations. (3)
Semidefinite relaxation within the causal space. We compare the performance of
PES with the recently developed Nevanlinna and Carath\'{e}odory continuation
methods and find that PES is more robust in the presence of noise and does not
require the usage of extended precision arithmetics. We also demonstrate that a
causal projection improves the performance of the Nevanlinna and
Carath\'{e}odory methods. The PES method is generalized to bosonic response
functions, for which the Nevanlinna and Carath\'{e}odory continuation methods
have not yet been developed. It is particularly useful for studying spectra
with sharp features, as they occur in the study of molecules and band
structures in solids.
