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We investigate the sets of uniform limits A ( B ‾ n ) , A ( D ‾ I ) of polynomials on the closed unit ball B ‾ n of C n and on the cartesian product D ‾ I where I is an arbitrary set, maybe finite, infinite denumerable or non-denumerable and D ‾ is the closed unit disc in C . The class A ( D ‾ I ) contains exactly all functions f : D ‾ I → C continuous with respect to the product topology on D ‾ I and separately holomorphic. We consider sets of uniqueness for A ( D ‾ I ) (respectively for A ( B ‾ n ) ) to be compact subsets K of T I (respectively of ∂ B ‾ n ) where T = ∂ D is the unit circle. If K has positive measure then K is a set of uniqueness. The converse does not hold. Finally, we do a similar study when the uniform convergence is not meant with respect to the usual Euclidean metric in C , but with respect to the chordal metric χ on C ∪ { ∞ } . [ABSTRACT FROM AUTHOR]

We prove concurrent universal Padé approximation for several universal Padé approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power series as well as in a subspace of A ∞. These results are valid for one center of expansion or for several centers as well. The Padé approximants allow generic approximation on arbitrary compact sets, not necessarily having connected component, in contrast with the partial sums of power series. We also establish affine genericity for a class of universal functions. [ABSTRACT FROM AUTHOR]