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  • 1.
    Coriasco, Sandro
    et al.
    Department of mathematics, Turin's university, Italy.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Global wave front set of modulation space typeManuscript (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.

  • 2.
    Coriasco, Sandro
    et al.
    Università degli Studi di Torino, Italy.
    Johansson, Karoline
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems2016In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, no 2, p. 285-333Article in journal (Refereed)
    Abstract [en]

    We illustrate the composition properties for an extended family of SG Fourier integral operators. We prove continuity results on modulation spaces, and study mapping properties of global wave-front sets for such operators. These extend classical results to more general situations. For example, there are no requirements on homogeneity for the phase functions. Finally, we apply our results to the study of the propagation of singularities, in the context of modulation spaces, for the solutions to the Cauchy problems for the corresponding linear hyperbolic operators. 

  • 3. Coriasco, Sandro
    et al.
    Johansson, Karoline
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Global wave-front sets of Banach, Fréchet and Modulation spacetypes, and pseudo-differential operators2013In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, no 8, p. 3228-3258Article in journal (Refereed)
    Abstract [en]

    We introduce global wave-front sets with respect to suitable Banach or Fréchet spaces. An important special case appears when choosing these spaces as modulation spaces. We show that the standard properties for known notions of wave-front set extend to global wave-front sets. In particular, we prove that microlocality and microellipticity hold for a class of globally defined pseudo-differential operators, acting continuously on the involved spaces.

  • 4.
    Coriasco, Sandro
    et al.
    Università degli Studi di Torino, Italy .
    Johansson, Karoline
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Global Wave-front Sets of Intersection and Union Type2014In: Fourier Analysis: Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations / [ed] Michael Ruzhansky, Ville Turunen, Heidelberg, New York, Dordrecht, London: Springer, 2014, p. 91-106Chapter in book (Refereed)
    Abstract [en]

    We show that a temperate distribution belongs to an ordered intersection or union of admissible Banach or Fréchet spaces if and only if the corresponding global wave-front set of union or intersection type is empty. We also discuss the situation where intersections and unions of sequences of spaces with two indices are involved. A main situation where the present theory applies is given by sequences of weighted, general modulation spaces.

  • 5.
    Coriasco, Sandro
    et al.
    Turin University, Italy.
    Johansson, Karoline
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Local wave-front sets of Banach and Fréchet types, and pseudo-differential operators2013In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 169, no 3-4, p. 285-316Article in journal (Refereed)
    Abstract [en]

    Let ω, ω 0 be appropriate weight functions and ${\fancyscript{B}}$ be an invariant BF-space. We introduce the wave-front set ${{\rm WF}_{\mathcal{B}}(f)}$ with respect to the weighted Fourier Banach space ${\mathcal{B}=\fancyscript{F} \fancyscript{B}(\omega )}$ . We prove that the usual mapping properties for pseudo-differential operators Op t (a) with symbols a in ${S^{(\omega_0)}_{\rho, 0}}$ hold for such wave-front sets. In particular we prove ${{\rm WF}_{\mathcal C}({\rm Op}_t (a) f)\subseteq {\rm WF}_{\mathcal{B}}(f)}$ and ${{\rm WF}_{\mathcal{B}}(f) \subseteq {\rm WF} _{\mathcal C}({\rm Op}_t (a) f)\bigcup {\rm Char} (a)}$ . Here ${\mathcal{C}=\fancyscript{F} \fancyscript{B}(\omega /\omega_0)}$ and Char(a) is the set of characteristic points of a.

  • 6.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    A counter example on nontangential convergence for oscillatory integrals2010In: Publications de l'Institut Mathématique (Beograd), ISSN 0350-1302, E-ISSN 1820-7405, Vol. 87, no 101, p. 129-137Article in journal (Refereed)
    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.

  • 7.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Association between temperate distributions and analytical functions in the context of wave-front sets2011In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 2, no 1, p. 65-89Article in journal (Refereed)
    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.

  • 8.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Generalized free time-dependent Schrödinger equation with initial data in Fourier Lebesgue spaces2011In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 2, no 4, p. 543-556Article in journal (Refereed)
    Abstract [en]

    Consider the solution of the free time-dependent Schrödinger equation with initial data f. It is shown by Sjögren and Sjölin that there exists f in the Sobolev spaceHs (Rd ), s = d/2 such that tangential convergence can not be widened to convergence regions. The author obtained in a previous paper the corresponding results for a generalized version of the Schrödinger equation, where −Δx is replaced by an operator ϕ(D), with special conditions on ϕ. In this paper we show that similar results may be obtained for initial data in usual and mixed Fourier Lebesgue spaces. We also relax the conditions on ϕ.

  • 9.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Propagation of singularities for pseudo-differential operators and generalized Schrödinger propagators2010Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis we discuss different types of regularity for distributions which appear in the theory of pseudo-differential operators and partial differential equations. Partial differential equations often appear in science and technology. For example the Schrödinger equation can be used to describe the change in time of quantum states of physical systems. Pseudo-differential operators can be used to solve partial differential equations.  They are also appropriate to use when modeling different types of problems within physics and engineering. For example, there is a natural connection between pseudo-differential operators and stationary and non-stationary filters in signal processing. Furthermore, the correspondence between symbols and operators when passing from classical mechanics to quantum mechanics essentially agrees with symbols and operators in the Weyl calculus of pseudo-differential operators.

    In this thesis we concentrate on investigating how regularity properties for solutions of partial differential equations are affected under the mapping of pseudo-differential operators, and in particular of the free time-dependent Schrödinger operators.

    The solution of the free time-dependent Schrödinger equation can be expressed as a pseudo-differential operator, with non-smooth symbol, acting on the initial condition. We generalize a result about non-tangential convergence, which was obtained by Sjögren and Sjölin (1989) for the free time-dependent Schrödinger equation.

    Another way to describe regularity for a distribution is to use wave-front sets. They do not only describe where the singularities are, but also the directions in which these singularities appear. The first types of wave-front sets (analytical wave-front sets) were introduced by Sato (1969, 1970). Later on Hörmander introduced ``classical'' wave-front sets (with respect to smoothness) and showed results in the context of pseudo-differential operators with smooth symbols, cf. Hörmander (1985).

    In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take B as a Banach space, which is invariant under translations and embedded between the space of Schwartz functions and the space of temperated distributions. Then we say that the wave-front set of a distribution contains all points (x0, ξ0) such that no localization of the distribution at x0, belongs to FB in the direction ξ0. We prove that pseudo-differential operators with smooth symbols shrink the wave-front set and we obtain opposite embeddings by using sets of characteristic points of the operator symbols.

  • 10.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Properties of wave-front sets and non-tangential convergence2011Doctoral 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.

  • 11.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Al-Talibi, Haidar
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Nyman, Peter
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Student's reasoning in the process of mathematical proofsManuscript (preprint) (Other academic)
    Abstract [en]

    This study focuses on students' way of reasoning about a proof in mathematics. The experiences of teaching students in the beginning of their studies at universities show that students have an obstacle in using deductive methods. The students' activity was designed specifically to investigate their deductive ability and to see if they can develop their way of reasoning. The group activities and interviews follow the students from the beginning where they, with great enthusiasm, begin colouring maps as a first sketch to a complete proof. The well-known statement to prove is chosen from a field in mathematics that the students are unfamiliar with, namely graph theory. More precisely it concerns the number of possible colourings of maps. Some university students have problems with constructing proofs, but in many cases the teacher can help them to reach a deductive reasoning.

  • 12.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Coriasco, Sandro
    University of Torino, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Local wave-front sets of Banach and Fréchet types, with applications to pseudo-differential operatorsManuscript (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.

  • 13.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Pilipovic, Stevan
    University of Novi Sad, Serbia .
    Teofanov, Nenad
    University of Novi Sad, Serbia .
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    A note on wave-front sets of Roumieu type ultradistributions2013In: Pseudo-Differential Operators, Generalized Functionsand Asymptotics / [ed] S. Molahajlo, S. Pilipovic, J. Toft, M. W. Wong, Basel Heidelberg NewYork Dordrecht London: Springer, 2013, p. 239-252Chapter in book (Refereed)
    Abstract [en]

    We study wave-front sets in weighted Fourier–Lebesgue spaces and corresponding spaces of ultradistributions. We give a comparison of these sets with the well-known wave-front sets of Roumieu type ultradistributions. Then we study convolution relations in the framework of ultradistributions. Finally, we introduce modulation spaces and corresponding wave-front sets, and establish invariance properties of such wave-front sets.

  • 14.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Pilipovic, Stevan
    Univ Novi Sad.
    Teofanov, Nenad
    Univ Novi Sad.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Gabor pairs, and a discrete approach to wave-front sets2012In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 166, no 2, p. 181-199Article in journal (Refereed)
    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 $f$.

  • 15.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Pilipovic, Stevan
    Teofanov, Nenad
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Micro-local analysis in some spaces of ultradistributions2012In: Publications de l'Institut Mathématique (Beograd), ISSN 0350-1302, E-ISSN 1820-7405, Vol. 92, no 106, p. 1-24Article in journal (Refereed)
    Abstract [en]

    We extend some results from [14] and [19], concerning wave-front sets of Fourier–Lebesgue and modulation space types, to a broader class of spaces of ultradistributions. We relate these wave-front sets one to another and to the usual wave-front sets of ultradistributions.

    Furthermore, we give a description of discrete wave-front sets by intro- ducing the notion of discretely regular points, and prove that these wave-front sets coincide with corresponding wave-front sets in [19]. Some of these inves- tigations are based on the properties of the Gabor frames.

  • 16.
    Johansson, Karoline
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Pilipovic, Stevan
    University of Novi Sad, Serbia.
    Teofanov, Nenad
    University of Novi Sad, Serbia.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Resolution of the wave- front set via discrete sets2013In: Proceedings in Applied Mathematics and Mechanics: PAMM, ISSN 1617-7061, E-ISSN 1617-7061, Vol. 13, p. 495-496Article in journal (Refereed)
    Abstract [en]

    We introduce discrete wave-front sets of sup type and prove that they coincide with the Hörmander wave-front set of a distribution. To that end we recall the notion of admissible lattice pairs and wave-front sets in Fourier-Lebesgue spaces.

  • 17.
    Toft, Joachim
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Johansson, Karoline
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Pilipovic, Stevan
    University of Novi Sad, Serbia.
    Teofanov, Nenad
    University of Novi Sad, Serbia.
    Sharp convolution and multiplication estimates in weighted spaces2015In: Analysis and Applications, ISSN 0219-5305, Vol. 13, no 5, p. 457-480Article in journal (Refereed)
    Abstract [en]

    We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations 248 (2010) 1972–2002].

1 - 17 of 17
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