EQUICONVERGENCE OF SOME SIMULTANEOUS HERMITE-PADE INTERPOLANTS

In several papers a result by J. L. Walsh on equiconvergence of polyno mial interpolation in the roots of unity to analytic functions, has be en extended using methods from complex analysis into the direction of rational interpolation to meromorphic functions having a given number of poles (E. B. Saff, A. Sharma and R. S. Varga; followed by M. P. Sto janova who introduced an extra integer parameter l greater than or equ al to 1 that governed the degree of the roots of unity in the first st age of the interpolation process, for l = infinity both stages use the same roots of unity). The aim of this paper is to extend the results indicated to the situation of simultaneous (or vector) rational interp olation to d-tuples of meromorphic functions, each analytic at the ori gin, having disjoint sets of poles of given (finite) cardinality; the main result exhibits so-called overconvergence: the difference between the rational interpolant for a fired l, 1 less than or equal to l < i nfinity and that for l = infinity converges to zero (geometrically) on a larger disk centered at the origin, than the disk of analyticity of the function that is interpolated.