Parton Distribution Functions (PDFs) grant us information on the distribution of quarks and gluons inside a hadron, in terms of fraction momentum carried by each parton. These functions are non-perturbative and are crucial in theoretical calculations describing high-energy hadron collisions. Despite their non-perturbative nature, the PDFs can be studied through experimental data and their Mellin moments. We present a novel approach that integrates these two sources of information, combining polynomial approximation techniques with the moment problem. This synergy allows to reconstruct distribution functions with a few integral constraints and few experimental data. Applying our novel methodology we reconstructed the scalar PDF $e^{q}(x)$ with limited experimental data and Mellin moments obtained from lattice QCD. We found functional forms for a flavor combination of $e^{q}(x)$ that satisfy the first two moments, among which the pion-nucleon sigma term, and closely approximate the provided data without requiring minimization techniques. These novel results allow us to improve our understanding of PDFs with limited access to Mellin moments and data points.