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  • 1.
    Abdeljawad, Ahmed
    et al.
    Univ Turin, Italy.
    Cappiello, Marco
    Univ Turin, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Pseudo-Differential Calculus in Anisotropic Gelfand-Shilov Setting2019In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 91, no 3, article id UNSP 26Article in journal (Refereed)
    Abstract [en]

    We study some classes of pseudo-differential operators with symbols a admitting anisotropic exponential type growth at infinity. We deduce mapping properties for these operators on Gelfand-Shilov spaces. Moreover, we deduce algebraic and certain invariance properties of these classes.

  • 2.
    Cappiello, Marco
    et al.
    Univ Turin, Italy.
    Rodino, Luigi
    Univ Turin, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the Inverse to the Harmonic Oscillator2015In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 40, no 6, p. 1096-1118Article in journal (Refereed)
    Abstract [en]

    Let b ( d ) be the Weyl symbol of the inverse to the harmonic oscillator on R- d . We prove that b ( d ) and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for b ( d ). In the even-dimensional case we characterize b ( d ) in terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.

  • 3.
    Cappiello, Marco
    et al.
    Univ Turin.
    Rodino, Luigi
    Univ Turin.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Radial symmetric elements and the Bargmann transform2014In: Integral transforms and special functions, ISSN 1065-2469, E-ISSN 1476-8291, Vol. 25, no 9, p. 756-764Article in journal (Refereed)
    Abstract [en]

    We prove that a function or distribution on R^d is radial symmetric, if and only if its Bargmann transform is a composition by an entire function on C and the canonical quadratic function from C^d to C. 

  • 4.
    Cappiello, Marco
    et al.
    University of Turin, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Pseudo-differential operators in a Gelfand–Shilov setting2017In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 290, no 5-6, p. 738-755Article in journal (Refereed)
    Abstract [en]

    We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes. 

  • 5.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Signahl, Mikael
    University of Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipovic' kernels2018In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 24, no 3, p. 666-698Article in journal (Refereed)
    Abstract [en]

    We prove that any linear operator with kernel in a Pilipovi{\'c} or Gelfand-Shilov spacecan be factorized by two operators in the same class. We also give links onnumerical approximations for such compositions. We apply these composition rulesto deduce estimates of singular values and establish Schatten-von Neumann propertiesfor such operators.

  • 6.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Signahl, Mikael
    University of Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Hilbert space embeddings for Gelfand–Shilov and Pilipović spaces2017In: Generalized Functions and Fourier Analysis: Dedicated to Stevan Pilipović on the Occasion of his 65th Birthday / [ed] Michael Oberguggenberger, Joachim Toft, Jasson Vindas, Patrik Wahlberg, Springer, 2017, p. 31-44Chapter in book (Refereed)
    Abstract [en]

    We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, H, and its dual, H′. We prove that for such quasi-Banach space B, there are convenient Hilbert spaces, Hk, k=1,2ss, with normalized Hermite functions as orthonormal bases and such that B lies between H1 and H1, and the latter spaces lie between H and H′.

  • 7.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Boundedness of Gevrey and Gelfand-Shilov kernels of positive semi-definite operators2015In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 6, no 2, p. 153-185Article in journal (Refereed)
    Abstract [en]

    We show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that positive elements with respect to the twisted convolution, belonging to a Gevrey class of certain order at the origin, belong to the Gelfand-Shilov space of the same order. In the end we apply these results to positive semi-definite pseudo-differential operators.

  • 8.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Wahlberg, Patrik
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    The Weyl product on quasi-Banach modulation spaces2019In: Bulletin of Mathematical Sciences, ISSN 1664-3607, E-ISSN 1664-3615, Vol. 9, no 2, p. 1-30, article id 1950018Article in journal (Refereed)
    Abstract [en]

    We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

  • 9.
    Cordero, Elena
    et al.
    Turins universitet.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Wahlberg, Patrik
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Sharp results for the Weyl product on modulation spaces2014In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 8, p. 3016-3057Article in journal (Refereed)
    Abstract [en]

    We give sufficient and necessary conditions on the Lebesgue exponentsfor the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to N=2 of aresult valid for the N-fold Weyl product. As a byproduct, we obtain sharpconditions for the twisted convolution to be bounded on Wieneramalgam spaces.

  • 10.
    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.

  • 11.
    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. 

  • 12. 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.

  • 13.
    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.

  • 14.
    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.

  • 15.
    Coriasco, Sandro
    et al.
    Università degli Studi di Torino, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    A calculus of Fourier integral operators with inhomogeneous phase functions on Rd2016In: Indian journal of pure and applied mathematics, ISSN 0019-5588, E-ISSN 0975-7465, Vol. 47, no 1, p. 125-166Article in journal (Refereed)
    Abstract [en]

    We construct a calculus for generalized SG Fourier integral operators, extending known results to a broader class of symbols of SG type. In particular, we do not require that the phase func- tions are homogeneous. An essential ingredient in the proofs is a general criterion for asymptotic expansions within the Weyl-Hörmander calculus. We also prove the L^2(R^d)-boundedness of the generalized SG Fourier integral operators having regular phase functions and amplitudes uni- formly bounded on R^{2d}.

  • 16.
    Coriasco, Sandro
    et al.
    Univ Turin, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Asymptotic expansions for Hörmander symbol classes in the calculus of pseudo-differential operators2014In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 5, no 1, p. 27-41Article in journal (Refereed)
    Abstract [en]

    We establish formulas for asymptotic expansions for S(m,g), the Hörmander class parameterized by the metric g and weight function m, defined on the phase space. By choosing m and g in appropriate ways, we cover some classical results on expansions for the standard symbol classes, and by choosing m and g in other ways we obtain asymptotic expansions for (generalized) SG classes.

  • 17.
    Fernandez, Carmen
    et al.
    Univ Valencia, Spain.
    Galbis, Antonio
    Univ Valencia, Spain.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Characterizations of GRS-weights, and consequences in time-frequency analysis2015In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 6, no 3, p. 383-390Article in journal (Refereed)
    Abstract [en]

    Let v be a submultiplicative weight. Then we prove that v satisfies Gel'fand-Raikov-Shilov-condition, if and only if is bounded for every positive . We use this equivalence to establish identification properties between weighted Lebesgue spaces, and between certain modulation spaces and Gelfand-Shilov spaces.

  • 18.
    Fernandez, Carmen
    et al.
    Universidad de Valencia.
    Galbis, Antonio
    Universidad de Valencia.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Spectral properties for matrix algebras2014In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 20, no 2, p. 362-383Article in journal (Refereed)
    Abstract [en]

    We consider Banach algebras of infinite matrices defined in terms of a weight measuring the off-diagonal decay of the matrix entries. If a given matrix A is invertible as an operator on l^2 we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in B(l^2), the Banach algebra of bounded operators on l^2. To this end we consider a condition on sequences of weights which extends the notion of GRS-condition. Finally we focus on the behavior of inverses of pseudodifferential operators whose Weyl symbols belong to weighted modulation spaces and the weights lack the GRS condition.

  • 19.
    Fernandez, Carmen
    et al.
    University of Valencia, Spain.
    Galbis Verdu, Antonio
    University of Valencia, Spain.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    The Bargmann transform and powers of harmonic oscillator on Gelfand-Shilov subspaces2017In: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, ISSN 1578-7303, Vol. 111, no 1, p. 1-13Article in journal (Refereed)
    Abstract [en]

    We consider the counter images H_♭(R^d) and H_{0,♭}(R^d) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipovic ́ spaces S_s(R^d ) and Σ_s(R^d) when 0 < s < 1/2 and deduce their images under the Bargmann transform.

  • 20. Gröchenig, Karl-Heinz
    et al.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces2011In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 114, no 1, p. 255-283Article in journal (Refereed)
    Abstract [en]

    We investigate the lifting property of modulation spaces and construct explicit isomorpisms between them. For each weight function \omega and suitable window function \varphi, the Toeplitz operator (or localization operator) TP _\varphi (\omega ) is an isomorphism from M^{p,q}_{(\omega _0)} into M^{p,q}_{(\omega _0/\omega )} for every p,q\in [1,\infty ] and arbitrary weight function \omega _0. The methods involve the pseudo-differential calculus of Bony and Chemin and the Wiener algebra property of certain symbol classes of pseudo-di®erential operators.

  • 21. Gröchenig, Karlheinz
    et al.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    The Range of Localization Operators and Lifting Theorems for Modulation and Bargmann-Fock Spaces2013In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 365, no 8, p. 4475-4496Article in journal (Refereed)
    Abstract [en]

    We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a refined version of the spectral invariance of pseudodifferential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.

  • 22.
    Holst, Anders
    et al.
    1.Department of Mathematics Lund University 221 00 Lund, Sweden.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Wahlberg, Patrik
    3.School of Electrical Engineering and Computer Science University of Newcastle Callaghan, NSW 2308, Australia.
    Weyl product algebras and classical modulation spaces2010In: Linear and non-linear theory of generalized functions and its applications / [ed] A. Kaminski, M. Oberguggenberger, S. Pilipovic, Warsaw: Polish Acad. Sci. Inst. Math. , 2010, p. 153-158Conference paper (Refereed)
    Abstract [en]

    We discuss continuity properties of the Weyl product when acting on classical modulation spaces. In particular, we prove that M p,q   is an algebra under the Weyl product when p∈[1,∞]  and 1≤q≤min(p,p ′ )  .

  • 23.
    Ivanenko, Yevhen
    et al.
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Custafsson, Mats
    Lund University.
    Jonsson, B. L. G.
    KTH Royal Instute of Technology.
    Luger, Annemarie
    Stockholm University.
    Nilsson, Börje
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Nordebo, Sven
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Passive Approximation and Optimization Using B-Splines2019In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 79, no 1, p. 436-458Article in journal (Refereed)
    Abstract [en]

    A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

  • 24.
    Ivanenko, Yevhen
    et al.
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Gustafsson, Mats
    Lund University.
    Jonsson, B.L.G.
    KTH Royal Institute of Technology.
    Luger, Annemarie
    Stockholm University.
    Nilsson, Börje
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Nordebo, Sven
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Passive approximation and optimization with B-splines2017Report (Other academic)
    Abstract [en]

    A passive approximation problem is formulated where the target function is an arbitrary complex valued continuous function defined on an approximation domain consisting of a closed interval of the real axis. The approximating function is any Herglotz function with a generating measure that is absolutely continuous with Hölder continuous density in an arbitrary neighborhood of the approximation domain. The norm used is induced by any of the standard Lp-norms where 1 ≤ p ≤ ∞. The problem of interest is to study the convergence properties of simple Herglotz functions where the generating measures are given by finite B-spline expansions, and where the real part of the approximating functions are obtained via the Hilbert transform. In practice, such approximations are readily obtained as the solution to a finite- dimensional convex optimization problem. A constructive convergence proof is given in the case with linear B-splines, which is valid for all Lp-norms with 1 ≤ p ≤ ∞. A number of useful analytical expressions are provided regarding general B-splines and their Hilbert transforms. A typical physical application example is given regarding the passive approximation of a linear system having metamaterial characteristics. Finally, the flexibility of the optimization approach is illustrated with an example concerning the estimation of dielectric material parameters based on given dispersion data. 

  • 25.
    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.

  • 26.
    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 $f$.

  • 27.
    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.

  • 28.
    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.

  • 29.
    Nordebo, Sven
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Gustafsson, Mats
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Nilsson, Börje
    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.
    Fisher information for inverse problems and trace class operators2012In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, no 12, p. Article ID: 123503-Article in journal (Refereed)
    Abstract [en]

    This paper provides a mathematical framework for Fisher information analysis forinverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, andthe Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriatespace for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictlysmaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information tobe trace class, a sufficient condition is formulated based on the singular values ofthe Jacobian as well as of the eigenvalues of the covariance operator, together withsome regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with “external”spherically isotropic noise, as well as “internal” additive uncorrelated noise.

  • 30.
    Nordebo, Sven
    et al.
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Gustafsson, Stefan
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Ioannidis, Andreas
    Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.
    Nilsson, Börje
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the generalized Jordan's lemma with applications in waveguide theory2013In: Proceeding of 2013 URSI International Symposium on Electromagnetic Theory (EMTS): International Conference Center Hiroshima, Hiroshima, Japan, May 20-24, 2013, 2013, p. 1039-1042Conference paper (Refereed)
    Abstract [en]

    This paper presents two variants of a generalized Jordan's lemma with applications in waveguide theory. As a main application is considered an asymptotic analysis for open waveguide structures with circular geometry. In particular, the generalized Jordan's lemma can be used to justify that field components can be calculated as the sum of discrete and non-discrete modes, i.e., as the sum of residues and an integral along the branch-cut defined by the transversal wavenumber of the exterior domain. An explicit example regarding the axial symmetric TM0 modes of a single core transmission line, wire, or optical fibre is included to demonstrate the associated asymptotic behavior for a typical open waveguide structure.

  • 31. Pilipovic, Stevan
    et al.
    Teofanov, Nenad
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Micro-Local Analysis in Fourier Lebesgue and modulation Spaces. Part I2011In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 17, no 3, p. 374-407Article in journal (Refereed)
    Abstract [en]

    Let ω,ω0 be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, of with respect to weighted Fourier Lebesgue space . We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in hold for such wave-front sets. Especially we prove that (*)Here Char (a) is the set of characteristic points of a.

  • 32.
    Pilipovic, Stevan
    et al.
    University of Novi Sad, Serbia .
    Teofanov, Nenad
    University of Novi Sad, Serbia .
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Singular Support and FL^q Continuity of Pseudodifferential Operators2011In: Approximation and Computation, Springer, 2011, p. 365-383Chapter in book (Refereed)
    Abstract [en]

    In this paper we show possible directions for numerical mathematicians interested in the approximation of different types of singular supports, wave front sets and of pseudodifferential operators in the framework of Fourier-Lebesgue spaces. The work contains new results on singular supports in Fourier-Lebesgue spaces and on the continuity properties of certain pseudodifferential operators.

  • 33. Pilipovic, Stevan
    et al.
    Teofanov, Nenad
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Wave-front sets in Fourier Lebesgue Spaces2008In: Rendiconti del seminario matematico, ISSN 0373-1243, Vol. 66, no 4, p. 299-320Article in journal (Refereed)
    Abstract [en]

    We consider wave-front sets in the framework of weighted Fourier Lebesgue spaces, FLqs. We prove that the usual mapping properties of pseudo-differential operators for such wave-front sets hold, especially for kernels which areproperly supported.

  • 34.
    Pilipovic, Stevan
    et al.
    University of Novi Sad, Serbien.
    Toft, JoachimLinnaeus University, Faculty of Technology, Department of Mathematics.
    Pseudo-Differential Operators and Generalized Functions2015Collection (editor) (Refereed)
    Abstract [en]

    At the ninth congress of the International Society for Analysis Applications and Computations (ISAAC), ISAAC Group in Pseudo-Differential Operators (IGPDO) and the ISAAC Group in Generalized Functions (IGGF) agreed to continue with the publication of a joint volume, as for the eight ISAAC Congress in Moscow 2013, of selecting papers from their two special sessions. Generalized functions as a general framework for almost all fields in analysis, and pseudo-differential operators as a basis of microlocal analysis in combination with harmonic and complex analysis with many applications, fit well and offer rich synergies for the further development of analysis in general.

    Moreover, the participants of both sessions agreed to dedicate this volume to Professor Michael Oberguggenberger at Insbruck university, Austria on his 60th birthday. Professor Oberguggenberger is one of the founder of the algebraic approach to generalized function theory with many contributions to the qualitative analysis of partial differential equations and a leader of the International Asso- ciation for Generalized Functions based in Vienna. Professor Oberguggenberger is highly appreciated as a scientist and a strong expert on generalized functions, especially Columbeau algebras. He is also very appreciated as a modest and encouraging person who supervised several PhD student to their examination. It is a pleasure for us to dedicate the volume to him.

    This joint volume is titled

    Pseudo-Differential Operators and Generalized Functions

    and consists of invited papers, mainly based on the scientific activities of the groups IGPDO and IGGF at the ninth ISAAC congress in Krakow, Poland, during August 2013. The volume is intended to be an independent sequel to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators”, “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” and “Pseudo-Differential Operators, Generalized Functions and Asymptotics”. These volumes were based on, respectively, the 4th ISAAC congress in Toronto in 2003, conference in V ̈axjo ̈ 2004, 5th ISAAC congress in Catana in 2005, 6th ISAAC congress in Ankara in 2007, workshop in Toronto in 2008, 7th ISAAC congress in London in 2009, and 8th ISAAC congress in Moscow ISAAC in 2011.

    The volume consists of 19 peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. Topics include algebras of generalized functions, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Variety of applications especially in the framework of manifolds with singular metrics and general relativity, microlocal analysis and the analysis of equations with singularities will be interested for a wide audience including graduate students and researchers in partial differential equations, mathematical physics, various fields of analysis, stochastic analysis and geometry.

    The papers can be sorted roughly into two groups. The first group of papers is related to generalized functions and deals with various problems of equations with singular coefficients and data within algebras of generalized functions where the classical method of regularizations is well established. Moreover in this setting, local analysis is well adapted and analyzed towards Höder type spaces and in the direction of generalized manifolds and applications in general relativity.

    In the second group of papers, various kinds of Fourier analysis are more present, involving micro-local analysis, harmonic analysis, theory of ultra-distributions, time-frequency analysis, etc. For example, Wiener type Tauberian theorems related to generalized integral transforms stochastic equations are adapted to classical distribution theory. Ultradistribution spaces are analyzed in connection with global type operators and wave fronts. Gabor analysis via modulation spaces or Hermite expansions is developed for various Gelfand-Shilov classes. Time-frequency methods are applied on evolution operators and on random MIMO systems. 

  • 35.
    Pilipovic, Stevan
    et al.
    Univ Novi Sad, Serbia.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Wave-front sets related to quasi-analytic Gevrey sequences2019In: Publications de l'Institut Mathématique (Beograd), ISSN 0350-1302, E-ISSN 1820-7405, Vol. 105, no 119, p. 1-16Article in journal (Refereed)
    Abstract [en]

    Quasi-analytic wave-front sets of distributions which correspond to the Gevrey sequence p!(s), s is an element of [1/2, 1) are defined and investigated. The propagation of singularities are deduced by considering sequences of Gaussian windowed short-time Fourier transforms of distributions which are modifications of the original distributions by suitable restriction-extension techniques. Basic micro-local properties of the new wave-fronts are thereafter established.

  • 36.
    Ruzhansky, Michael
    et al.
    Imperial College London, UK.
    Sugimoto, Mitsuru
    Nagoya University, Japan.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Tomita, Naohito
    Osaka University, Japan.
    Changes of variables in modulation and Wiener amalgam spaces2011In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 284, no 16, p. 2078-2092Article in journal (Refereed)
    Abstract [en]

    In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of such spaces and, as a consequence, we obtain several versions of local and global Beurling–Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on FLq. Finally, counterparts of these results are discussed for spaces on the torus.

  • 37.
    Signahl, Mikael
    et al.
    Univ Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Mapping properties for the Bargmann transform on modulation spaces2012In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 3, no 1, p. 1-30Article in journal (Refereed)
    Abstract [en]

    We investigate the mapping properties for the Bargmann transform and prove that this transform is isometricand bijective from modulation spaces to convenient Banach spaces of analytic functions.

  • 38. Signahl, Mikael
    et al.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Remarks on mapping properties for the Bargmann transform on modulation spaces2011In: Integral transforms and special functions, ISSN 1065-2469, E-ISSN 1476-8291, Vol. 22, no 4-5, p. 359-366Article in journal (Refereed)
    Abstract [en]

    We investigate the mapping properties for the Bargmann transform and prove that this transform is isometricand bijective from modulation spaces to convenient Banach spaces of analytic functions.

  • 39.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes2017In: Analysis and Applications, ISSN 0219-5305, Vol. 15, no 3, p. 353-389Article in journal (Refereed)
    Abstract [en]

    We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in (0, ∞]. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate H ̈ormander classes. 

  • 40.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Continuity and Schatten Properties for Pseudo-Differential Operators on Modulation Spaces2007In: Modern Trends in Pseudo-Differential Operators, Birkhäuser, Basel-Boston-Berlin , 2007, p. 173-206Chapter in book (Refereed)
  • 41.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces2007In: Modern Trends in Pseudo-Differential Operators, Birkhäuser, Basel-Boston-Berlin , 2007, p. 313-328Chapter in book (Refereed)
  • 42.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Continuity of Gevrey-Hörmander pseudo-differential operators on modulation spaces2019In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 10, no 2, p. 337-358Article in journal (Refereed)
    Abstract [en]

    Let s ≥ 1, ω,ω ∈ P^0_{E,s} , a ∈ 􏰒\Gamma _s^(ω_0), and let B be a suitable invariant quasi-Banach function space. Then we prove that the pseudo-differential operator Op(a) is continuous from M(ω_0·ω, B) to M(ω, B).

  • 43.
    Toft, Joachim
    Blekinge Tekniska Högskola.
    Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I2004In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 207, no 2, p. 399-429Article in journal (Refereed)
    Abstract [en]

    Let Mp,q denote the modulation space with parameters p,q[1,∞]. If 1/p1+1/p2=1+1/p0 and 1/q1+1/q2=1/q0, then it is proved that Mp1,q1*Mp2,q2Mp0,q0. The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.

  • 44.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Continuity Properties for Modulation Spaces, with Applications to Pseudo-Differential Calculus, II2004In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 26, no 1, p. 73-106Article in journal (Refereed)
    Abstract [en]

    We discuss continuity for weighted modulation spaces, andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces. We also discussthe trace operatoraa(0, ·) acting on modulationspaces. The results are used to get inclusions betweenmodulation spaces and Besov spaces, and proving continuityfor pseudo-differential operators and Toeplitz operators.

  • 45.
    Toft, Joachim
    Blekinge Tekniska Högskola.
    Continuity properties for non-commutative convolution algebras with applications in pseudo-differential calculus2002In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 126, no 2, p. 115-142Article in journal (Refereed)
    Abstract [en]

    We study continuity properties for a family {sp}p1 of increasing Banach algebras under the twisted convolution, which also satisfies that asp, if and only if the Weyl operator aw(x,D) is a Schatten–von Neumann operator of order p on L2. We discuss inclusion relations between the sp-spaces, Besov spaces and Sobolev spaces. We prove also a Young type result on sp for dilated convolution. As an application we prove that f(a)s1, when as1 and f is an entire odd function. We finally apply the results on Toeplitz operators and prove that we may extend the definition for such operators.

  • 46.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces2005In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 84, no 3, p. 269-282Article in journal (Refereed)
    Abstract [en]

    For any appropriate weight function ω, spaces are introduced as counterimage of unweighted modulation spaces through Toeplitz operators with symbol ω. It is proved that the weighted modulation spaces coincide with them when ω is a suitable hypoelliptic symbol. It is furhter proved that a necessary and sufficient condition for the embedding   between two modulation spaces to be compact is that the quotient ω21 vanishes at infinity.

  • 47.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
    Fourier modulation spaces and positivity in twisted convolution algebra2006In: Integral Transforms and Special Functions, ISSN 1065-2469, Vol. 17, no 02-03, p. 193-198Article in journal (Refereed)
  • 48.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Gabor Analysis for a Broad Class of Quasi-Banach Modulation Spaces2015In: Pseudo-Differential Operators and Generalized Functions / [ed] Stevan Pilipovic, Joachim Toft, Springer, 2015, p. 255-284Chapter in book (Refereed)
    Abstract [en]

    We extend the Gabor analysis in [13] to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norms. For such spaces we deduce invariance and embedding properties, and that the elements admit reconstructible sequence space representations using Gabor frames. We apply these results to show identies between sets of compactly supported elements in modulation spaces and Fourier Lebesgue spaces. 

  • 49.
    Toft, Joachim
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Hudson's Theorem and Rank One Operators in Weyl Calculus2006In: Pseudo-Differential Operators and Related Topics, Birkhäuser, Basel-Boston-Berlin , 2006Chapter in book (Refereed)
  • 50.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Images of function and distribution spaces under the Bargmann transform2017In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 8, no 1, p. 83-139Article in journal (Refereed)
    Abstract [en]

    Weconsiderabroadfamilyoftestfunctionspacesandtheirdual(distribu- tion) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic ́. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions. 

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