ON THE DEGREE OF NONUNIQUENESS OF SIMULTANEOUS HERMITE-PADE APPROXIMANTS TO A VECTOR OF ENTIRE-FUNCTIONS

Let (f1, f2, ..., f(m)) be a vector of formal power series, let (rho0, rho1, ..., rho(m)) be an (m + 1)-tuple of non-negative integers,and let sigma := rho0 + rho1 + rho2 + ... + rho(m). The simultaneous Hermite-Pade approximation problem involves looking for polynomials P1, P2, ..., P(m) and Q, with deg(Q) less-than-or-equal-to sigma - rho0; deg(P(j)) less-than-or-equal-to sigma - rho(j), 1 less-than-or-equal-to j less-than-or-equal-to m such that (f(j)Q - P(j))(z) = O(z(sigma+1)), 1 less-than-or-equal-to j less-than-or-equal-to m. It is well known, and easy to see, that there is a solution to this problem. In this paper, we consider uniqueness of the vector (P1/Q,P2/Q, ..., P(m)/Q) of rational functions, which may be regarded as simultaneous Hermite-Pade approximants to (f1, f2, ..., f(m)). For special classes of functions, uniqueness has been proved, but the uniqueness problem has not been solved in general. We show that even when f1, f2, ..., f(m) are ''independent'' in a strong sense, there may be uncountably many distinct vectors (P1/Q, P2/Q, ..., P(m)/Q) solving the problem. Moreover, the degree of non-uniqueness is the same as in the problem to find Q of degree less-than-or-equal-to sigma - rho0, and a P1 of degree less-than-or-equal-to sigma - rho1 satisfying (f1Q - P1)(z) = O(z(sigma + 1)). When compared to the classical Pade problem (f1Q - P1)(z) = O(z(sigma-rho0)+(sigma-rho1)+1), this indicates that the number of degrees of freedom, or non-uniqueness, is sigma - (rho0 + rho1).