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  • 1.
    Asekritova, Irina
    Växjö universitet, Matematiska och systemtekniska institutionen.
    Interpolation of Approximation Spaces with Nonlinear Projectors2006In: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 55, no 3, p. 146-149Article in journal (Refereed)
    Abstract [en]

    Approximation spaces defined by multiparametric approximation families with possible nonlinear projectors are considered. It is shown that a real interpolation space for a tuple of such spaces is again an approximation space of the same type.

  • 2.
    Asekritova, Irina
    Växjö universitet, Matematiska och systemtekniska institutionen.
    Regularization Theory and Real Interpolation2008Conference paper (Other academic)
    Abstract [en]

    In inverse problems we often need to solve numerically unstable problems that are very sensitive to noise. One of the approaches to such type of problems is classical regularization theory for Hilbert spaces. In the talk I plan to show connections between this theory and the theory of real interpolation, give di®erent examples and discuss a non - Hilbert case - the couple (L2;BV ).

  • 3.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Brudnyi, Yuri
    Technion, Haifa, Israel.
    Interpolation of Multiparameter Approximation Spaces2004In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 129, no 2, p. 182-206Article in journal (Refereed)
    Abstract [en]

    We prove a general interpolation theorem for linear operators acting simultaneously in several approximation spaces which are defined by multiparametric approximation families. As a consequence, we obtain interpolation results for finite families of Besov spaces of various types including those determined by a given set of mixed differences.

  • 4.
    Asekritova, Irina
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Cobos, Fernando
    Complutense University of Madrid.
    Kruglyak, Natan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Interpolation of Closed Subspaces and Invertibility of Operators2015In: Journal of Analysis and its Applications, ISSN 0232-2064, Vol. 34, no 2015, p. 1-15Article in journal (Refereed)
    Abstract [en]

    Let (Y0,Y1) be a Banach couple and let Xj be a closed complemented subspace of Yj, (j = 0,1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X0,X1)θ,q is a closed subspace of (Y0,Y1)θ,q. In particular, we establish conditions which are necessary and sufficient for the equality (X0,X1)θ,q = (Y0,Y1)θ,q, with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X1 = Y1 and X0 is a subspace of codimension one in Y0. 

  • 5.
    Asekritova, Irina
    et al.
    School of Mathematics and System Engineering, Växjö University, Sweden.
    Kruglyak, Natan
    Department of Mathematics, Luleå University of Technology, Sweden.
    Interpolation of Besov Spaces in the Non-Diagonal Case2007In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 18, no 4, p. 511-516Article in journal (Refereed)
    Abstract [en]

    In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points includes a ball of , we have where

    and

    .

  • 6.
    Asekritova, Irina
    et al.
    Växjö universitet, Sweden.
    Kruglyak, Natan
    Luleå tekniska universitet, Sweden.
    Invertibility of Operators in Spaces of Real Interpolation2008In: Revista Matemática Complutense, ISSN 1139-1138, Vol. 21, no 1, p. 207-217Article in journal (Refereed)
    Abstract [en]

    Let be a linear bounded operator from a couple to a couple such that the restrictions of on the spaces and have bounded inverses. This condition does not imply that the restriction of on the real interpolation space has a bounded inverse for all values of the parameters and . In this paper under some conditions on the kernel of we describe all spaces such that the operator has a bounded inverse.

  • 7.
    Asekritova, Irina
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kruglyak, Natan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Necessary and sufficient conditions for invertibility of operators in spaces of real interpolation2013In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 264, no 1, p. 207-245Article in journal (Refereed)
    Abstract [en]

    Let A be a bounded linear operator from a couple (X-0, X-1) to a couple (Y-0, Y-1) such that the restrictions of A on the end spaces X-0 and X-1 have bounded inverses defined on Y-0 and Y-1, respectively. We are interested in the problem of how to determine if the restriction of A on the space (X-0, XI)(theta,q) has a bounded inverse defined on the space (Y-0, Y-1)(theta,q). In this paper, we show that a solution to this problem can be given in terms of indices of two subspaces of the kernel of the operator A on the space X-0 + X-1.

  • 8.
    Asekritova, Irina
    et al.
    Department of Mathemathics, Yaroslavl' Pedagogical University, Russia.
    Kruglyak, Natan
    Department of Mathematics, Yaroslavl' State University, Russia.
    On Equivalemce of K- and J-Methods for (n+1)-Tuples of Banach Spaces1997In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 122, no 2, p. 99-116Article in journal (Refereed)
    Abstract [en]

    It is shown that the main results of the theory of real interpolation, i.e. the equivalence and reiteration theorems, can be extended from couples to a class of (n+1)-tuples of Banach spaces, which includes (n+1)-tuples of Banach function lattices, Sobolev and Besov spaces. As an application of our results, it is shown that Lions' problem on interpolation of subspaces and Semenov's problem on interpolation of subcouples have positive solutions when all spaces are Banach function lattices or their retracts. In general, these problems have negative solutions.

  • 9.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Kruglyak, Natan
    Department of Mathematics, Lulea University of Technology, Sweden.
    Real Interpolation of Vector-Valued Spaces in Non-Diagonal Case2004In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 133, no 6, p. 1665-1675Article in journal (Refereed)
    Abstract [en]

    It is shown that the formula

    where and is correct under the restrictions and It is also true if we suppose that and the spaces are functional Banach or quasi-Banach lattices on the same measure space

  • 10.
    Asekritova, Irina
    et al.
    Växjö University.
    Kruglyak, Natan
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    The Besicovitch covering theorem and near-minimizers for the couple (L-2, BV)2010In: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, no 1, p. 29-33Article in journal (Refereed)
    Abstract [en]

    Let Omega be a rectangle in R-2. A new algorithm for the construction of a near-minimizer for the couple (L-2 (Omega); BV(Omega)) is presented. The algorithm is based on the Besicovitch covering theorem and analysis of local approximations of the given function f is an element of L-2 (Omega).

  • 11.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Kruglyak, Natan
    Department of Mathematics, Luleå University of Technology, Sweden.
    The Besikovitch Covering Theorem and Near Minimizers for the Couple (L2,BV)2010In: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 59, no 1, p. 29-33Article in journal (Refereed)
    Abstract [en]

    Let Ω be a rectangle in R2. A new algorithm for the construction of a near-minimizer for the couple (L2(Ω), BV(Ω)) is presented. The algorithm is based on the Besicovitch covering theorem and analysis of local approximations of the given function f ∈ L2(Ω).

  • 12.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Kruglyak, Natan
    Department of Mathematics, Yaroslavl' State University, Russia.
    Maligranda, Lech
    Department of Mathematics, Luleå University of Technology, Sweden.
    Persson, Lars-Erik
    Department of Mathematics, Luleå University of Technology, Sweden.
    Distribution and Rearranement Estimates of the Maximal Functions and Interpolation1997In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 124, no 2, p. 107-132Article in journal (Refereed)
    Abstract [en]

    There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.

  • 13.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Kruglyak, Natan
    Department of Mathematics Luleå University of Technology, Luleå, Sweden.
    Nikolova, Ludmila
    University of Sofia, Bulgaria.
    Lizorkin-Freitag Formula for Several Weighted Lp Spaces and Vector-Valued Interpolation2005In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 170, no 3, p. 227-239Article in journal (Refereed)
    Abstract [en]

    A complete description of the real interpolation space L=(Lp0(ω0),…,Lpn(ωn))θ⃗ ,q is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces Ωi (i∈I) such that L is an lq sum of the restrictions of L to Ωi, and L on each Ωi is a result of interpolation of just two weighted Lp spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

  • 14.
    Asekritova, Irina
    et al.
    Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM.
    Nikolova, Ludmila
    Sofia University, Sofia, Bulgaria.
    Kruglyak, Natan
    Luleå University of Technology, Luleå, Sweden.
    Maligranda, Lech
    Luleå University of Technology, Luleå, Sweden.
    Persson, Lars-Erik
    Luleå University of Technology, Luleå, Sweden.
    Lions-Peetre Reiteration Formulas for Triples and Their Application2001In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 145, no 3, p. 219-254Article in journal (Refereed)
    Abstract [en]

    We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted Lp-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability in the frame of Besov spaces based on Lorentz spaces. Moreover, by using the results and ideas of this paper, we can extend the Stein–Weiss interpolation theorem known for Lp(μ)-spaces with change of measures to Lorentz spaces with change of measures. In particular, the results obtained show that for some problems in analysis the three-space real interpolation approach is really more useful than the usual real interpolation between couples.

  • 15.
    Asekritova, Irina
    et al.
    Växjö universitet, Sweden.
    Nilsson, Börje
    Växjö universitet, Sweden.
    Rydström, Sara
    Växjö universitet, Sweden.
    Diffractive Index Determination by Tikhonov Regularization on Forced String Vibration Data2009In: Mathematical modeling of wave phenomena, American Institute of Physics (AIP), 2009, p. 224-232Conference paper (Other academic)
    Abstract [en]

    Wave analysis is efficient for investigating the interior of objects. Examples are ultra sound examination of humans and radar using elastic and electromagnetic waves. A common procedure is inverse scattering where both transmitters and receivers are located outside the object or on its boundary. A variant is when both transmitters and receivers are located on the scattering object. The canonical model is a finite inhomogeneous string driven by a harmonic point force. The inverse problem for the determination of the diffractive index of the string is studied. This study is a first step to the problem for the determination of the mechanical strength of wooden logs. An inverse scattering theory is formulated incorporating two regularizing strategies. The results of simulations using this theory show that the suggested method works quite well and that the regularization methods based on the couple of spaces (L2; H1 ) could be very useful in such problems.

1 - 15 of 15
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