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

Jiménez-Garrido, Javier; Sanz, Javier; Schindl, Gerhard

doi:10.1007/s00605-019-01345-y

Cited by 13 publications

(47 citation statements)

References 19 publications

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“…Finally, condition (M, N ) γ r has appeared in the recent work by the authors [13] (mainly again for r ∈ N >0 ). There we have generalized the results from [31] to the auxiliary ultradifferentiable-like function classes, moreover in [13, Theorem 5.9], we have given a generalization of the ultradifferentiable Whitney extension results from [7] involving a ramification parameter r > 0.…”

Section: The Indices (M N) and ( !)mentioning

confidence: 84%

“…The motivation of defining the mixed growth indices (especially for weight functions) was arising by proving [13,Theorem 5.9]. First we wish to mention that in [7,Commentaires 32] it was observed (without giving a proof) that there is a connection between (M, N ) γ 1 and (σ, ω) γ 1 (under suitable basic assumptions on M, N ): They have stated that (M, N ) γ 1 does imply (ω M , ω N ) γ 1 and so generalizing [15,Prop.…”

Section: Remarkmentioning

confidence: 99%

“…In the weight sequence case we refer to [7] for the Whitney jet mapping and to [31] for the Borel mapping, in the weight function case see [5] for the Borel mapping and [17,24,25] for the general Whitney jet mapping. In our recent paper [13], which has served as motivation for this article, by involving a ramification parameter r ∈ N >0 we have generalized the mixed setting results from [31] to r -ramification classes introduced in [30]. We have also generalized the Whitney extension results from [7] by using a parameter r > 0 (see [13,Theorem 5.9]).…”

Section: Introductionmentioning

confidence: 99%

“…In our recent paper [13], which has served as motivation for this article, by involving a ramification parameter r ∈ N >0 we have generalized the mixed setting results from [31] to r -ramification classes introduced in [30]. We have also generalized the Whitney extension results from [7] by using a parameter r > 0 (see [13,Theorem 5.9]).…”

Section: Introductionmentioning

confidence: 99%

“…Then a similar notion is required in the mixed setting to obtain satisfactory theorems. Therefore, motivated by the occurring mixed ramified conditions between M and N and their associated weight functions ω M and ω N appearing in [13] the definition of the mixed growth index for sequences γ (M, N ) and for weight functions γ (σ, ω) has been given, see Sect. 3.1.…”

Section: Introductionmentioning

confidence: 99%

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Ultraholomorphic extension theorems in the mixed setting

Jiménez-Garrido

Sanz

Schindl

2020

Banach J. Math. Anal.

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The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important. KeywordsSpaces of ultraholomorphic functions • Weight sequences and weight functions • Growth indices • (non)quasianalyticity of function classes • Restriction and extension mappings • Surjectivity of the Borel map Mathematics Subject Classification 26E10 • 30D60 • 46A13 • 46E10 Communicated by Mieczyslaw Mastylo.

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Section: The Indices (M N) and ( !)mentioning

confidence: 84%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ultraholomorphic extension theorems in the mixed setting

Jiménez-Garrido

Sanz

Schindl

2020

Banach J. Math. Anal.

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Construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

Boiti,

Jornet,

Oliaro

et al. 2024

Mathematische Nachrichten

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We give a simple construction of the log‐convex minorant of a sequence and consequently extend to the ‐dimensional case the well‐known formula that relates a log‐convex sequence to its associated function , that is, . We show that in the more dimensional anisotropic case the classical log‐convex condition is not sufficient: convexity as a function of more variables is needed (not only coordinate‐wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.

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On the Inclusion Relations of Global Ultradifferentiable Classes Defined by Weight Matrices

Boiti,

Jornet,

Oliaro

et al. 2024

Mediterr. J. Math.

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We study and characterize the inclusion relations of global classes in the general weight matrix framework in terms of growth relations for the defining weight matrices. We consider the Roumieu and Beurling cases, and as a particular case, we also treat the classical weight function and weight sequence cases. Moreover, we construct a weight sequence which is oscillating around any weight sequence which satisfies some minimal conditions and, in particular, around the critical weight sequence $$(p!)^{1/2}$$ ( p ! ) 1 / 2 , related with the non-triviality of the classes. Finally, we also obtain comparison results both on classes defined by weight functions that can be defined by weight sequences and conversely.

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

Cited by 13 publications

References 19 publications

Ultraholomorphic extension theorems in the mixed setting

Ultraholomorphic extension theorems in the mixed setting

Construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

On the Inclusion Relations of Global Ultradifferentiable Classes Defined by Weight Matrices

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