A linear Tschebyshev-Pade (Frobenius-Pade) approximant to a function expanded in orthonormal polynomials on [-1,1] can be rewritten exactly as the ratio -Q_{n,1}/Q_{n,2} of type I Hermite-Pade polynomials for the tuple [1, f1, f2] whose pair f1, f2 forms a Nikishin system. Learners see how this identity converts a convergence question about free-pole rational approximation of an orthogonal expansion into a question about the limiting zero distribution of Hermite-Pade polynomials, which is then attacked through multipoint Pade interpolation, the S-property, and a vector equilibrium problem for a pair of measures lifted to the two-sheeted Riemann surface of sqrt(z^2-1). The programme extends Stahl's theory for multivalued functions to the Tschebyshev-Pade setting and remains conjectural.
A Tschebyshev-Pade (linear Frobenius-Pade) approximant of order n to a function f expanded in polynomials orthonormal on the interval [-1,1] with respect to a measure mu is a ratio P_n/Q_n of polynom…