Kinematic model-independent reconstruction of Palatini $f(R)$ cosmology
  A kinematic treatment to trace out the form of $f(R)$ cosmology, within the
Palatini formalism, is discussed by only postulating the universe homogeneity
and isotropy. To figure this out we build model-independent approximations of
the luminosity distance through rational expansions. These approximants extend
the Taylor convergence radii computed for usual cosmographic series. We thus
consider both Pad\'e and the rational Chebyshev polynomials. They can be used
to accurately describe the universe late-time expansion history, providing
further information on the thermal properties of all effective cosmic fluids
entering the energy momentum tensor of Palatini's gravity. To perform our
numerical analysis, we relate the Palatini's Ricci scalar with the Hubble
parameter $H$ and thus we write down a single differential equation in terms of
the redshift $z$. Therefore, to bound $f(R)$, we make use of the most recent
outcomes over the cosmographic parameters obtained from combined data surveys.
In particular our clue is to select two scenarios, i.e. $(2,2)$ Pad\'e and
$(2,1)$ Chebyshev approximations, since they well approximate the luminosity
distance at the lowest possible order. We find that best analytical matches to
the numerical solutions lead to $f(R)=a+bR^n$ with free parameters given by the
set $(a, b, n)=(-1.627, 0.866, 1.074)$ for $(2,2)$ Pad\'e approximation,
whereas $f(R)=\alpha+\beta R^m$ with $(\alpha, \beta, m)=(-1.332, 0.749,
1.124)$ for $(2,1)$ rational Chebyshev approximation. Finally, our results are
compared with the $\Lambda$CDM predictions and with previous studies in the
literature. Slight departures from General Relativity are also discussed.
