Shohat–Favard type theorem for orthogonal series

A function f is considered in the form of its formal series expansion ∑ k = 0 ∞ f k P k where ( P k ) k = 0 ∞ denotes a given system of orthogonal polynomials with respect to a measure on the real line. We prove that the denominators, numerators and residuals of their Frobenius–Padé approximation satisfy a five-term recurrence relation. The converse result is obtained: starting from the sequence of denominators, the function f is reconstructed, i.e. the analogous of the classical Shohat–Favard theorem is done.