Bernstein Polynomials based Probabilistic Interpretation of Quark Hadron Duality

It is now widely recognized that large Bjorken $x$ data play an important role in global analyses of Parton Distribution Functions (PDFs) even at collider energies, through perturbative QCD evolution. For values of the scale of the reaction, $Q^2$, in the multi-GeV region the structure functions at large $x$ present resonance structure. Notwithstanding, these data can be incorporated in the analyses by using quark-hadron duality or approximate scaling of the structure function data averaged over their resonance structure. Several averaging methods have been proposed using either the PDFs Mellin moments, or their truncated moments. We propose an alternative method using Bernstein polynomials integrals, or Bernstein moments. Bernstein moments render a smooth form of the structure function in the resonance region. Furthermore, being based on a different averaging criterion than the methods adopted so far, they provide a new framework for understanding the possible mechanisms giving origin to the phenomenon of quark-hadron duality.