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Properties of wave-front sets and non-tangential convergence
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
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 [en]
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: urn:nbn:se:lnu:diva-12963ISBN: 978-91-86491-96-3 (print)OAI: oai:DiVA.org:lnu-12963DiVA: diva2:429609
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
List of papers
1. Local wave-front sets of Banach and Fréchet types, with applications to pseudo-differential operators
Open this publication in new window or tab >>Local wave-front sets of Banach and Fréchet types, with applications to pseudo-differential operators
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let ω, ω0 be appropriate weight functions and B be an invariant BF-space. We introduce the wave-front set WFFB(ω)(f) with respect to weighted Fourier Banach space FB(ω). We prove the usual mapping properties for pseudo-differential operators Opt(a) with symbols a inS^{ω0}_{ρ,0} hold for such wave-front sets.

Keyword
Wave-front, Fourier, Banach, modulation, micro-local
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-2437 (URN)
Note
Pre-print in ArXiv:0911.1867Available from: 2010-04-14 Created: 2010-04-13 Last updated: 2012-01-03Bibliographically approved
2. Global wave front set of modulation space type
Open this publication in new window or tab >>Global wave front set of modulation space type
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce global wave-front sets WFB(f), f in S'(Rd), with respect to suitable Banach or Fréchet spaces B. An important special case is given by the modulation spaces B=M(ω,B), where ω is an appropriate weight function and B is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to WFB(f). In particular, we prove that microlocality and microellipticity hold for a class of globally defined pseudo-differential operators Opt(a), acting continuouslyon the involved spaces.

Keyword
Wave-front, Fourier, Banach, modulation, micro-local
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-12959 (URN)
Available from: 2011-06-23 Created: 2011-06-23 Last updated: 2017-01-11Bibliographically approved
3. Gabor pairs, and a discrete approach to wave-front sets
Open this publication in new window or tab >>Gabor pairs, and a discrete approach to wave-front sets
2012 (English)In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 166, no 2, 181-199 p.Article in journal (Refereed) Published
Abstract [en]

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of "continuous type". This implies that the coefficients of a Gabor frame expansion of $f$ are parameter dependent, and describe the wave-front set of $f$.

Place, publisher, year, edition, pages
Springer, 2012
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-10207 (URN)10.1007/s00605-011-0288-2 (DOI)
Projects
matematisk modellering
Available from: 2011-01-19 Created: 2011-01-19 Last updated: 2017-01-11Bibliographically approved
4. Association between temperate distributions and analytical functions in the context of wave-front sets
Open this publication in new window or tab >>Association between temperate distributions and analytical functions in the context of wave-front sets
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
Keyword
Analytic function, Wave-front set, Fourier Banach function space
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-11101 (URN)10.1007/s11868-011-0022-9 (DOI)
Available from: 2011-03-15 Created: 2011-03-12 Last updated: 2013-04-10Bibliographically approved
5. A counter example on nontangential convergence for oscillatory integrals
Open this publication in new window or tab >>A counter example on nontangential convergence for oscillatory integrals
2010 (English)In: Publications de l'Institut Mathématique (Beograd), ISSN 0350-1302, E-ISSN 1820-7405, Vol. 87, no 101, 129-137 p.Article in journal (Refereed) Published
Abstract [en]

Consider the solution of the time-dependent Schrödinger equation with initial data f. It is shown by Sjögren and Sjölin (1989) that there exists f in the Sobolev space Hs(Rn), s=n/2 such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when -Δx is replaced by an operator φ(D), with special conditions on φ.

Place, publisher, year, edition, pages
Beograd: Mathematical Institute of the Serbian Academy of Sciences and Arts, 2010
Keyword
Generalized time-dependent Schrödinger equation, nontangential convergence
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-2438 (URN)
Available from: 2010-04-14 Created: 2010-04-13 Last updated: 2017-03-08Bibliographically approved

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