lnu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Association between temperate distributions and analytical functions in the context of wave-front sets
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
2011 (English)In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, Vol. 2, no 1, 65-89 p.Article in journal (Refereed) Published
Abstract [en]

Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Ω = {z in Cd; | Im z| < 1 } such that the wave-front set of f of Fourier BF-space types in intersection with Rd ×Sd-1 consists of the points (x, ξ) such that F does not belong to the Fourier BF-space at xi ξ.

Place, publisher, year, edition, pages
Basel: Birkhäuser , 2011. Vol. 2, no 1, 65-89 p.
Keyword [en]
Analytic function, Wave-front set, Fourier Banach function space
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-11101DOI: 10.1007/s11868-011-0022-9OAI: oai:DiVA.org:lnu-11101DiVA: diva2:403763
Available from: 2011-03-15 Created: 2011-03-12 Last updated: 2013-04-10Bibliographically approved
In thesis
1. Properties of wave-front sets and non-tangential convergence
Open this publication in new window or tab >>Properties of wave-front sets and non-tangential convergence
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider regularity properties for solutions to partial differential equations and pseudo-differential equations. The thesis mainly concerns wave-front sets and micro-local properties. Regularity properties are also viewed in terms of nontangential convergence for the generalized free time-dependent Schrödinger equations, where the Laplace operator is replaced by more general functions.

Wave-front sets describe location of singularities and the directions of their propagation. We establish usual and convenient mapping properties for such wave-front sets under action of pseudodifferential operators with smooth symbols.

We define three components of wave-front sets with respect to appropriate Banach and Fréchet spaces, in order to describe local properties as well as behavior far away, including heavy oscillations. The union of these components is called the global wavefront set. For these wave-front sets, we establish micro-local and micro-ellipticity properties for pseudo-differential operators in appropriate symbol classes. We obtain the classical wave-front sets as special cases (cf. Hörmander [9]). For the type of wave-front sets which describe local properties we also prove equivalence between wave-front sets of Fourier Banach function and modulation space types.

To open up for numerical computations we introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier Lebesgue and modulation spaces. Furthermore, we prove that these wave-front sets agree with each other and with the corresponding wave-front sets of continuous type. We also consider the link between analytic functions and temperate distributions in terms of such wave-front sets.

The last part of this thesis concerns counter examples of nontangential convergence for the generalized time-dependent Schrödinger equation with initial data in Sobolev spaces.

Place, publisher, year, edition, pages
Växjö, Kalmar: Linnaeus University Press, 2011. 161 p.
Series
Linnaeus University Dissertations, 58/2011
Keyword
Fourier Banach spaces, Generalized free time-dependent Schrödinger equation, Micro-local, Modulation spaces, Non-tangential convergence, Pseudo-differential operators, Regularity, Wave-front sets.
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-12963 (URN)978-91-86491-96-3 (ISBN)
Public defence
2011-10-20, Weber, Universitetsplatsen 1, Växjö, 13:15 (English)
Opponent
Supervisors
Available from: 2011-08-09 Created: 2011-06-23 Last updated: 2017-01-11Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text

Search in DiVA

By author/editor
Johansson, Karoline
By organisation
School of Computer Science, Physics and Mathematics
In the same journal
Journal of Pseudo-Differential Operators and Applications
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar

Altmetric score

Total: 59 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf