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In the last few decades, extensions of General Relativity have reached alwaysmore attention especially in view of possible breakdowns of the standard$\Lambda$CDM paradigm at intermediate and high redshift regimes. If GeneralRelativity would not be the ultimate theory of gravity, modifying Einstein'sgravity in the homogeneous and isotropic universe may likely represent a viablepath toward the description of current universe acceleration. We here focus ourattention on two classes of extended theories, i.e. the $f(R)$ and$f(R,G)$-gravity. We parameterize the so-obtained Hubble function by means ofeffective barotropic fluids, by calibrating the shapes of our curves throughsome of the most suitable dark energy parameterizations, XCDM, CPL, WP.Afterwards, by virtue of the correspondence between the Ricci scalar and theGauss-Bonnet topological invariant with the redshift $z$, we rewrite $f(R,G)$in terms of corresponding $f(z)$ auxiliary functions. This scheme enables oneto get numerical shapes for $f(R,G)$ and $f(R)$ models, through acoarse-grained inverse scattering procedure. Although our procedure agrees withthe simplest extensions of general relativity, it leaves open the possibilitythat the most suitable forms of $f(R)$ and $f(R,G)$ are rational Pad\'epolynomials of first orders. These approximations seem to be compatible withnumerical reconstructions within intermediate redshift domains and match fairlywell small redshift tests.

Kinematic reconstructions of extended theories of gravity at small and intermediate redshifts