Projektdetails
Abstract
Wider research context / theoretical framework
Spaces of ultradifferentiable functions are subclasses of smooth functions; classically defined in terms of weight sequences or functions assuming standard growth assumptions on the weights. Using a weight matrix one is able to treat both cases in a unified approach but also more classes. Ultraholomorphic
classes are their complex counterparts; in the literature mostly defined in terms of weight sequences. Very recently we have also introduced ultraholomorphic classes defined by weight functions and matrices.
Hypotheses/research questions /objectives
This project is devoted to questions concerning the injectivity and surjectivity of the (asymptotic) Borel map considered on classes of ultradiff. and ultraholom. functions, both of Roumieu- and Beurling-type and defined in terms of weight sequences, functions and most generally matrices. In the case when the Borel
map is not surjective the aim is to get information or even a full characterization on sequences belonging to the image. In the ultradiff. setting, for the more general Whitney jet mapping and in the Beurling case for classes defined by weight functions and matrices, we are treating the problem when it is not surjective but a controlled loss of regularity is possible (mixed setting). Finally, we want to study the structure of ultraholom. classes in more detail and find new applications for spaces defined in terms of weight functions and weight matrices in order to underline the autonomy and importance of these new classes.
Approach/methods
Studying these questions we are using methods from Complex and Functional Analysis and we are working with weight matrices to consider a general setting and treat both the sequence and function case simultaneously.
Level of originality / innovation
The results obtained within this project will complete and refine the information in the literature for ultraholom. classes defined by weight sequences but also provide completely new insights for classes defined by weight functions. By working with matrices we will obtain automatically results for mixed weight
sequence situations (controlled loss of regularity) and working with weaker assumptions on the weights than usually assumed in the weight sequence approach. The analogous behaviour of ultraholom. classes compared with the ultradiff. setting (e.g. characterization of inclusion relations) will be investigated,
applications for ultraholom. classes defined by weight functions will be given to emphasize the autonomy of this new setting and finally we will obtain information about the failure of the surjectivity of the Borel and Whitney jet mapping in the ultradiff. setting.
Primary researchers involved
On this project will work the applicant Gerhard Schindl with a Postdoc at the University of Vienna and his collaboration partners Javier Sanz Gil from the Universidad de Valldadolid, Céline Esser from the Université de Liège and Armin Rainer from the University of Vienna.
Spaces of ultradifferentiable functions are subclasses of smooth functions; classically defined in terms of weight sequences or functions assuming standard growth assumptions on the weights. Using a weight matrix one is able to treat both cases in a unified approach but also more classes. Ultraholomorphic
classes are their complex counterparts; in the literature mostly defined in terms of weight sequences. Very recently we have also introduced ultraholomorphic classes defined by weight functions and matrices.
Hypotheses/research questions /objectives
This project is devoted to questions concerning the injectivity and surjectivity of the (asymptotic) Borel map considered on classes of ultradiff. and ultraholom. functions, both of Roumieu- and Beurling-type and defined in terms of weight sequences, functions and most generally matrices. In the case when the Borel
map is not surjective the aim is to get information or even a full characterization on sequences belonging to the image. In the ultradiff. setting, for the more general Whitney jet mapping and in the Beurling case for classes defined by weight functions and matrices, we are treating the problem when it is not surjective but a controlled loss of regularity is possible (mixed setting). Finally, we want to study the structure of ultraholom. classes in more detail and find new applications for spaces defined in terms of weight functions and weight matrices in order to underline the autonomy and importance of these new classes.
Approach/methods
Studying these questions we are using methods from Complex and Functional Analysis and we are working with weight matrices to consider a general setting and treat both the sequence and function case simultaneously.
Level of originality / innovation
The results obtained within this project will complete and refine the information in the literature for ultraholom. classes defined by weight sequences but also provide completely new insights for classes defined by weight functions. By working with matrices we will obtain automatically results for mixed weight
sequence situations (controlled loss of regularity) and working with weaker assumptions on the weights than usually assumed in the weight sequence approach. The analogous behaviour of ultraholom. classes compared with the ultradiff. setting (e.g. characterization of inclusion relations) will be investigated,
applications for ultraholom. classes defined by weight functions will be given to emphasize the autonomy of this new setting and finally we will obtain information about the failure of the surjectivity of the Borel and Whitney jet mapping in the ultradiff. setting.
Primary researchers involved
On this project will work the applicant Gerhard Schindl with a Postdoc at the University of Vienna and his collaboration partners Javier Sanz Gil from the Universidad de Valldadolid, Céline Esser from the Université de Liège and Armin Rainer from the University of Vienna.
| Status | Abgeschlossen |
|---|---|
| Tatsächlicher Beginn/ -es Ende | 1/09/20 → 30/04/24 |