Cosmographic analysis with Chebyshev polynomials
  The limits of standard cosmography are here revised addressing the problem of
error propagation during statistical analyses. To do so, we propose the use of
Chebyshev polynomials to parameterize cosmic distances. In particular, we
demonstrate that building up rational Chebyshev polynomials significantly
reduces error propagations with respect to standard Taylor series. This
technique provides unbiased estimations of the cosmographic parameters and
performs significatively better than previous numerical approximations. To
figure this out, we compare rational Chebyshev polynomials with Pad\'e series.
In addition, we theoretically evaluate the convergence radius of (1,1)
Chebyshev rational polynomial and we compare it with the convergence radii of
Taylor and Pad\'e approximations. We thus focus on regions in which convergence
of Chebyshev rational functions is better than standard approaches. With this
recipe, as high-redshift data are employed, rational Chebyshev polynomials
remain highly stable and enable one to derive highly accurate analytical
approximations of Hubble's rate in terms of the cosmographic series. Finally,
we check our theoretical predictions by setting bounds on cosmographic
parameters through Monte Carlo integration techniques, based on the
Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic
data, using the JLA supernovae sample and the most recent versions of Hubble
parameter and baryon acoustic oscillation measurements. We find that
cosmography with Taylor series fails to be predictive with the aforementioned
data sets, while turns out to be much more stable using the Chebyshev approach.
