On the maximal extension in the mixed ultradifferentiable weight sequence setting

Abstract

For the ultradifferentiable weight sequence setting it is known that the Borel map which assigns to each function the infinite jet of derivatives (at 0) is surjective onto the corresponding weighted sequence class if and only if the sequence is strongly nonquasianalytic for both the Roumieu-and Beurling-type classes. Sequences which are nonquasianalytic but not strongly nonquasianalytic admit a controlled loss of regularity and we determine the maximal sequence for which such a mixed setting is possible for both types, hence get information on the controlled loss of surjectivity in this situation. Moreover, we compare the optimal sequences for both mixed strong nonquasianalyticity conditions arising in the literature.

On optimal solutions of the Borel problem in the Roumieu case

“…cf. [10,Lemma 5.8] and [24,Remark 2.1]. So ≺ γ1 induces a relation on equivalence classes of positive sequences which is always antisymmetric and becomes transitive if we restrict to weight sequences of moderate growth.…”

“…If M ′ is a suitable other weight sequence, then Λ {M ′ } ⊆ j ∞ E {M} (R) is equivalent to a condition (namely (4.1)) purely in terms of M ′ , M which we denote by M ′ ≺ SV M ; see [26]. There is an explicit positive sequence L = L(M ) such that L ≺ SV M , that is Λ {L} ⊆ j ∞ E {M} (R), and L is optimal with respect to this property; see [24]. (2) The condition M ′ ≺ γ1 M (see (4.4)) is generally stronger than M ′ ≺ SV M ; it plays a crucial role in the more general Whitney problem [5,7,13,20,18].…”

Sectorial extensions for ultraholomorphic classes defined by weight functions

“…see [13,Sect. 5] and [9, Lemma 4.1], [21,Lemma 2.8] and [7,Lemma 2.4] and the references mentioned in the proofs there.…”

Untitled

“…1/j = +∞, see e.g. [17,Lemma 2.8] and [7,Lemma 2.4] and the references mentioned in the proofs there. Moreover, lim j→+∞ (M j /j!)…”

“…For M ∈ LC it is known that non-quasianalyticity implies lim j→+∞ µj j = +∞; see e.g. [17,Prop. 4.4].…”