Flat functions in Carleman ultraholomorphic classes via proximate orders

Abstract

Whenever the defining sequence of a Carleman ultraholomorphic class (in the sense of H. Komatsu) is strongly regular and associated with a proximate order, flat functions are constructed in the class on sectors of optimal opening. As consequences, we obtain analogues of both Borel-Ritt-Gevrey theorem and Watson's lemma in this general situation.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1402.166

The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces

“…An important question in both the ultradifferentiable and ultraholomorphic situation is to establish sufficient and necessary conditions on M under which the Borel map B 0 , which assigns to f the infinite jet (f (j) (0)) j∈N , is onto the corresponding sequence spaces Λ {M} or Λ (M) (defined in Subsection 2.9), see [14], [24] and [19]. In the ultradifferentiable setting the so-called strong non-quasianalyticity condition (γ 1 ) is characterizing this behavior for both types as shown in [14].…”

Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders

“…As it was proved in [48,Th. 3.4], or as it can be deduced from the theory of O-regular variation (see [22,Remark 2.1.19]), for every strongly regular sequence M one has ω(M) ∈ (0, ∞).…”

“…All the weight sequences admitting a nonzero proximate order are strongly regular, what shows that some of the hypotheses in [48,30] Although not every strongly regular sequence admits a nonzero proximate order, as shown in [24,Example 4.16], admissibility holds true for every strongly regular sequence appearing in applications.…”

“…The existence or not of nontrivial flat functions was characterized in [48,Coro. 4.12] for classes defined from a strongly regular sequence M (see Definition 2.2) such that the function d M is a nonzero proximate order (Definition 2.7).…”

“…Furthermore, the function U satisfies, in its domain, the properties (I)-(VI) in Theorem 2.9 of the functions of the class M F (ργ, ρ * (s)). [30] about M−summability were initially stated for strongly regular sequences M such that d M is a proximate order (a fortiori a nonzero proximate order, as deduced from [48,Th. 3.4]), since then Maergoiz functions were available.…”