Universal Padé approximants of Seleznev type

Abstract

The theory of universal Taylor series can be extended to the case of Padé approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Padé approximants of some power series. We present the first generic result in this direction, for Padé approximants corresponding to Taylor developments of holomorphic functions in simply connected domains. The universal approximation is required only on compact sets K which lie outside the domain of definition and have connected complement. If the sets K are additionally disjoint from the boundary of the domain of definition, then the universal functions can be smooth on the boundary. Subject Classification MSC2010 : primary 41A21, 30K05 secondary 30B10, 30E10, 30K99, 41A10, 41A20.

Generic properties of Padé approximants and Padé universal series

“…Upon performing a simple division, we are then lead to Padé universal series. Together with Baire category theorem, this allows us to recover the fact that U S (D) is residual in H(D), [14,Theorem 3.1].…”

“…Proof. This is nothing but Baire Category Theorem together with [14,Theorem 3.1], since N m,n is an open dense subset of H(D).…”

“…Theorem 5.8 (Theorem 3.1 of [14]). Let S = (p m , q m ) m be a sequence of pairs of positive integers such that (p n ) n is unbounded.…”

“…Several articles recently dealt with the notion of Padé universal series, an analog of universal series, where the role of partial Taylor sums is played by Padé approximants, see e.g. [14,15,28]. For a power series f at the origin, we denote by [f ; p n /q n ] the Padé approximant of degree (p n , q n ) to f , see Section 2 for more details on Padé approximants.…”

“…i) [f ; p n /q n ] converges to h in A(K) along the subsequence S ′ ; ii) [f ; p n /q n ] converges to f in H(Ω) along the subsequence S ′ . Assuming that the sequence (p n ) n is unbounded, the set U S (Ω) of Padé universal series is residual in H(Ω), see [14,Theorem 3.1]. Actually, it is easy to see, by a Rouché type argument in the complement of Ω, that the hypothesis on the sequence (p n ) n is mandatory for that result to hold true.…”