Universal Taylor series on products of planar domains

Partially Smooth Universal Taylor Series on products of simply connected domains

“…Remark 3.6 In the extreme case where S i = ∅ for all i, or S i = ∂ Ω for all i, we get the results of [9], regarding holomorphic and smooth, respectively, universal Taylor series.…”

“…The only difference occurs when Theorem Remark 3.10 We notice that the previous theroems have been proved with a specific enumeration of the monomials in mind. The question whether there exist similar results simultaneously with respect to all enumerations of the monomials has been answered in the negative in [9], Theorem 6.1, for the case where S i = ∅ for all i = 1, 2, . .…”

“…d i=1 K i , where K i are planar compact sets with connected complements and f : K → C is a continuous function such that f • Φ is holomorphic on the disk D ⊂ C, for every injective holomorphic mapping Φ : D → K, then f is uniformly approximable on K by polynomials. This result was recently used in[9], where it is proved that there exist holomorphic functions ond i=1 Ω i , where Ω i are simply connected domains of C, that behave universally on all products of planar compact sets, with connected complements, disjoint from d i=1 Ω i , and also that there exist smooth functions on d i=1 Ω i , where Ω i are simply connected domains of C, such that C − Ω i is connected, which behave universally on all products of planar compact sets, with connected complements, disjoint from d i=1 Ω i . In the present paper, these…”

Some remarks on approximation in several complex variables

“…Thus polynomials are uniformly dense in 𝐴 𝐷 (𝐾). See also[1,6,7].Thus, 𝑃(𝐾) = 𝐴(𝐾) is equivalent to 𝑃(𝐾) = 𝐴 𝐷 (𝐾) and 𝐴 𝐷 (𝐾) = 𝐴(𝐾). A combination of Theorem 2.1 and Corollary 3.1 completes the proof.…”