  Simulations of finite temperature quantum systems provide imaginary frequency
Green's functions that correspond one-to-one to experimentally measurable
real-frequency spectral functions. However, due to the bad conditioning of the
continuation transform from imaginary to real frequencies, established methods
tend to either wash out spectral features at high frequencies or produce
spectral functions with unphysical negative parts. Here, we show that
explicitly respecting the analytic `Nevanlinna' structure of the Green's
function leads to intrinsically positive and normalized spectral functions, and
we present a continued fraction expansion that yields all possible functions
consistent with the analytic structure. Application to synthetic trial data
shows that sharp, smooth, and multi-peak data is resolved accurately.
Application to the band structure of silicon demonstrates that high energy
features are resolved precisely. Continuations in a realistic correlated setup
reveal additional features that were previously unresolved. By substantially
increasing the resolution of real frequency calculations our work overcomes one
of the main limitations of finite-temperature quantum simulations.
