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  • 1
    Electronic Resource
    Electronic Resource
    J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations (1997)
    Springer
    Integral equations and operator theory 29 (1997), S. 373-454 
    Details
    ISSN: 1420-8989
    Keywords: 30E05 ; 30D99 ; 34A55 ; 34L30 ; 47A56 ; 47A57
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is the first of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted largely to foundational material (much of which is of independent interest) on the theory of assorted classes of meromorphic matrix valued functions. Particular attention is paid to the structure of J-inner functions and connections with bitangential interpolation problems and reproducing kernel Hilbert spaces. Some new characterizations of regular, singular and strongly regular J-inner functions in terms of the associated reproducing kernel Hilbert spaces are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01193811
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  • 2
    Electronic Resource
    Electronic Resource
    J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem (2000)
    Springer
    Integral equations and operator theory 36 (2000), S. 11-70 
    Details
    ISSN: 1420-8989
    Keywords: 30E05 ; 30D99 ; 34A55 ; 34L40 ; 47A56 ; 47A57
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01236286
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  • 3
    Electronic Resource
    Electronic Resource
    J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem (2000)
    Springer
    Integral equations and operator theory 36 (2000), S. 127-181 
    Details
    ISSN: 1420-8989
    Keywords: 30E05 ; 30D99 ; 34A55 ; 34L40 ; 47A56 ; 47A57
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is a continuation of our study of the inverse monodromy problem for canonical systems of integral and differential equations which appeared in a recent issue of this journal. That paper established a parametrization of the set of all solutions to the inverse monodromy for canonical integral systems in terms of two continuous chains of matrix valued inner functions in the special case that the monodromy matrix was strongly regular (and the signature matrixJ was not definite). The correspondence between the chains and the solutions of this monodromy problem is one to one and onto. In this paper we study the solutions of this inverse problem for several different classes of chains which are specified by imposing assorted growth conditions and symmetries on the monodromy matrix and/or the matrizant (i.e., the fundamental solution) of the underlying equation. These external conditions serve to either fix or limit the class of admissible chains without computing them explicitly. This is useful because typically the chains are not easily accessible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202092
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  • 4
    Electronic Resource
    Electronic Resource
    Trace formulas for pair operators (1978)
    Springer
    Integral equations and operator theory 1 (1978), S. 152-175 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = χ(x)gv(x−y)+[1−χ(x)]fv(x−y) to the square |x|,|y| ⩽ T and shall evaluate the limit as T ↑ ∞ . Here χ denotes the indicator function of the right half-line [0,∞) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01690981
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  • 5
    Electronic Resource
    Electronic Resource
    On the inverse spectral problem for the string equation (1978)
    Springer
    Integral equations and operator theory 1 (1978), S. 270-277 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The theory of Hilbert spaces of entire functions is used to deduce the changes in the mass distribution of a string effected by a small change in its spectral function. The formula of Gelfand-Levitan emerges as a special case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01690986
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  • 6
    Electronic Resource
    Electronic Resource
    More on maximum entropy interpolants and maximum determinant completions of associated Pick matrices (1996)
    Springer
    Integral equations and operator theory 24 (1996), S. 188-229 
    Details
    ISSN: 1420-8989
    Keywords: Primary 47A57 ; Secondary 30E05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01193460
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  • 7
    Electronic Resource
    Electronic Resource
    On three Krein extension problems and some generalizations (1998)
    Springer
    Integral equations and operator theory 31 (1998), S. 1-91 
    Details
    ISSN: 1420-8989
    Keywords: 30E05 ; 42A82 ; 47A56 ; 47A57
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01203457
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  • 8
    Electronic Resource
    Electronic Resource
    On an extension problem, generalized fourier analysis, and an entropy formula (1980)
    Springer
    Integral equations and operator theory 3 (1980), S. 143-215 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let h(t) be an n × n matrix valued function on the interval |t| ⩽ τ with summable entries. Let ĥ denote the Fourier transform of h and let e denote the n × n identity matrix. Necessary and sufficient conditions for the existence of an extension u of h to the full line such that e-û admits either a left or a right canonical factorization and the inverse transform of (e-û)−1-e vanishes for |t| ⩾ τ are presented and discussed. The connections between these extensions and a generalized Fourier transform are then explored in detail with the help of the theory of triangular factorization. It is then shown that if an allied finite Wiener-Hopf operator based on h is positive, then h admits exactly one extension of the type alluded to above. This extension is then characterized in terms of an entropy integral.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01682992
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  • 9
    Electronic Resource
    Electronic Resource
    Hilbert spaces of analytic functions, inverse scattering and operator models.II (1985)
    Springer
    Integral equations and operator theory 8 (1985), S. 145-180 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first, in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric]operators with finite and equal deficiency indices. A number of other related issues and applications are then discussed briefly. These include entropy inequalities, interpolation, parametrization ofJ inner matrices, the Schur algorithm and canonical equations. Finally, a list of misprints for Part I is incorporated at the end.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202812
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  • 10
    Electronic Resource
    Electronic Resource
    On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy (1989)
    Springer
    Integral equations and operator theory 12 (1989), S. 757-812 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Reproducing kernel space methods are used to study the truncated matrix Hamburger moment problem on the line, an associated interpolation problem and the maximum entropy solution. Enroute a number of formulas are developed for orthogonal matrix polynomials associated with a block Hankel matrix (based on the specified matrix moments for the Hamburger problem) under less restrictive conditions than positive definiteness. An analogue of a recent formula of Alpay-Gohberg and Gohberg-Lerer for the number of roots of certain associated matrix polynomials is also established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01196878
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    Paper (German National Licenses)
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