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We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products K = ∏ i = 1 d K i , where K i ⊆ C are compact sets and C \ K i are connected, i = 1 , ... , d and 0 ∉ K . On such K the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraically generic. [ABSTRACT FROM AUTHOR]

Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains Ω i , i = 1 , ... , d . The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets K i , i = 1 , ... , d such that C - K i is connected and for at least one i 0 the set K i 0 is disjoint from Ω i 0 . [ABSTRACT FROM AUTHOR]

We show generic existence of power series a = ∑ n = 0 ∞ a n z n , a n ∈ C , such that the sequence T N (a) (z) = ∑ n = 0 N b n (a 0 , ... , a n) z n , N = 0 , 1 , 2 ... approximates every polynomial uniformly on every compact set K ⊂ C \ { 0 } with connected complement. The functions b n : C n + 1 → C are assumed to be continuous and such that for every a 0 , a 1 , ... , a n - 1 ∈ C , the function C ∋ z → b n (a 0 , a 1 , ... , a n - 1 , z) is onto C . This clearly covers the case of linear functions b n : b n (a 0 , ... , a n) = ∑ k = 0 n λ n , k a k , λ n , k ∈ C , λ n , n ≠ 0 . [ABSTRACT FROM AUTHOR]

Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains Ω i that extend continuously on ∏ i = 1 d (Ω i ∪ S i) , where S i are subsets of ∂ Ω i , open in the relative topology. The universal approximation occurs on every product of compact sets K i such that C - K i are connected and for some i 0 it holds K i 0 ∩ (Ω i 0 ∪ S i 0 ¯) = ∅ . [ABSTRACT FROM AUTHOR]

V. Nestoridis conjectured that if Ω is a simply connected domain of C that does not contain 0 and S (Ω) is the set of all functions f ∈ H (Ω) with the property that the set { T n (f) (z) = ∑ k = 0 n f (k) (z) / k ! (− z) k : n = 0 , 1 , 2 , ... } is dense in H (Ω) , then S (Ω) is a dense G δ set in H (Ω). We answer the conjecture in the affirmative in the special case where Ω is an open disc D (z 0 , r) that does not contain 0. [ABSTRACT FROM AUTHOR]

In this article we deal with the hypercyclic complex Taylor shift operator acting on the space of holomorphic functions in a suitable simply connected domain, along with the real Taylor shift on the space of real infinitely differentiable functions. In both cases, we investigate a problem of interpolation for the class of hypercyclic elements. Further, we study the notion of disjoint hypercyclicity for subsequences of the orbit of the real Taylor shift. [ABSTRACT FROM AUTHOR]

Let be countably many increasing sequences of integers tending to ∞. For a power series , we denote by the nth partial sum of f. Under some condition on , we construct lacunary power series converging in the unit disk , with arbitrarily large lacunes, which satisfy the following property (P): for any countable family of compact sets in the complement of , each of them with connected complement, the set is dense in , endowed with the product topology, inherited from the -norm of the space of all continuous functions on K which are holomorphic in its interior. The set of such countably universal series is shown to be invariant under some summability processes. Our construction also allows us to exhibit power series with Padé approximants enjoying countably universal properties, and to prove that the set of all power series with radius of convergence 1 which satisfy (P) is either void or spaceable. We finally use Ostrowski-gaps to give a very direct proof of two known results: (1) The set of doubly universal Taylor series is non-void only if for some increasing sequence , and (2) The set of frequently universal series is void. [ABSTRACT FROM AUTHOR]

Abstract In this note, the universality of a sequence of operators associated to the partial sums of the Taylor series of a holomorphic function is investigated. The emphasis is put on the fact that the Taylor series are evaluated at a prescribed point and the variable is the center of the expansion. The dynamics of the sequence of operators linked to the partial sums of a power series that is not generated by an entire function is also studied. [ABSTRACT FROM AUTHOR]

Let ( λ n ) be a strictly increasing sequence of positive integers. Inspired by the notions of topological multiple recurrence and disjointness in dynamical systems, Costakis and Tsirivas have recently established that there exist power series ∑ k ≥ 0 a k z k with radius of convergence 1 such that the pairs of partial sums { ( ∑ k = 0 n a k z k , ∑ k = 0 λ n a k z k ) : n = 1 , 2 , … } approximate all pairs of polynomials uniformly on compact subsets K ⊂ { z ∈ C : | z | ≥ 1 } , with connected complement, if and only if lim sup n λ n n = + ∞ . In the present paper, we give a new proof of this statement avoiding the use of tools of potential theory. It allows to study the case of doubly universal infinitely differentiable functions. We obtain also the algebraic genericity of the set of such power series. Further we show that the Cesàro means of partial sums of power series with radius of convergence 1 cannot be frequently universal. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition so that we have disjoint universality for sequences of operators that map a holomorphic function to a partial sum of its Taylor expansion. This problem is connected with doubly universal Taylor series and this is an effort to generalize the concept to multiply universal Taylor series. [ABSTRACT FROM AUTHOR]

We establish properties concerning the distribution of poles of Padé approximants, which are generic in Baire category sense. We also investigate Padé universal series, an analog of classical universal series, where Taylor partial sums are replaced with Padé approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infinite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Padé universal series. We also show how Padé universal series can be built from classical universal series with large Ostrowski-gaps. [ABSTRACT FROM AUTHOR]

We introduce classes of universal Taylor series, both topologically and algebraically generic, whose image under some regular matrix summability methods are automatically universal. [ABSTRACT FROM AUTHOR]

Abstract: For a holomorphic function in the unit disk, denotes the -th partial sum of the Taylor development of with center at . We show that given a strictly increasing sequence of positive integers , there exists a holomorphic function on the unit disk such that the pairs of partial sums approximate all plausibly approximable functions uniformly on suitable compact subsets of the complex plane if and only if . This provides a new strong notion of universality for Taylor series. [Copyright &y& Elsevier]

Abstract: In the present article we study an interpolation problem for classes of analytic functions, in a systematic manner. More precisely, we provide sufficient conditions so that proper and “big”, in the Baire category sense, subclasses of analytic functions have an interpolation property at an infinite set of points. We then apply our main theorems to several classes of universal, hypercyclic functions. [Copyright &y& Elsevier]

For a holomorphic function f in the open unit disc D and ζ ∈ D , S n (f , ζ) denotes the n -th partial sum of the Taylor development of f at ζ. Given an increasing sequence of positive integers μ = (μ n) , we consider the classes U (D , ζ) and U (μ) (D , ζ) of holomorphic functions f in D such that subsequences of the partial sums { S n (f , ζ) : n = 1 , 2 , ... } and { S μ n (f , ζ) : n = 1 , 2 , ... } respectively approximate all polynomials uniformly on the compact sets K ⊂ { z ∈ C : | z | ≥ 1 } with connected complement. We show that these two classes of universal Taylor series coincide if and only if lim sup n → + ∞ ( μ n + 1 μ n ) < + ∞. In the same spirit, we prove that, for ζ ≠ 0 , the equality U (μ) (D , ζ) = U (μ) (D , 0) holds if and only if lim sup n → + ∞ ( μ n + 1 μ n ) < + ∞. Finally we deal with the case of real universal Taylor series. [ABSTRACT FROM AUTHOR]