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" Multivariate prediction, de Branges spaces, and related extension and inverse problems / "
Damir Z. Arov, Harry Dym.
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BL
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| Record Number
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864173
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| Main Entry
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Arov, Damir Z.
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| Title & Author
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Multivariate prediction, de Branges spaces, and related extension and inverse problems /\ Damir Z. Arov, Harry Dym.
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| Publication Statement
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Cham, Switzerland :: Birkhäuser,, 2018.
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| Series Statement
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Operator theory: advances and applications ;; volume 266
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1 online resource
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| ISBN
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3319702629
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: 9783319702629
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3319702610
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9783319702612
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; Chapter 1 Introduction; 1.1 Organization of the monograph; 1.2 Notation; 1.3 de Branges matrices E and de Branges spaces B(E); 1.4 Some basic identifications; 1.5 Direct and inverse spectral problems; 1.6 Jp-inner mvf 's and de Branges matrices; 1.7 Helical extension problems; 1.8 Positive extension problems; 1.9 Accelerant extension problems; 1.10 Inverse spectral problems for Krein systems; 1.11 Prediction for multivariable processes based on a finite segment of the past; Weakly stationary processes; Processes with weakly stationary increments.
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2.5 The Stieltjes class2.6 The classes Gp×p∞ (0) and Gp×pa (0); 2.7 The classes Pp×p∞ and Pp×pa; 2.8 The classes ˚ Ap×p∞ and ˚ Ap×p; 2.9 Supplementary notes; Chapter 3 The de Branges Spaces B(E) and H(A); 3.1 Reproducing kernel Hilbert spaces; 3.2 Entire de Branges matrices E and the spaces B(E); 3.3 A characterization of B(E) spaces; 3.4 Connections between E ∈ I(jp) andA ∈ U(Jp); 3.5 The RKHS H(A) and its connection with B(E); 3.6 Closed R0-invariant subspaces of H(A) and B(E); 3.7 Supplementary notes; Chapter 4 Three Extension Problems; 4.1 The helical extension problem
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| Abstract
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This monograph deals primarily with the prediction of vector valued stochastic processes that are either weakly stationary, or have weakly stationary increments, from finite segments of their past. The main focus is on the analytic counterpart of these problems, which amounts to computing projections onto subspaces of a Hilbert space of p x 1 vector valued functions with an inner product that is defined in terms of the p x p matrix valued spectral density of the process. The strategy is to identify these subspaces as vector valued de Branges spaces and then to express projections in terms of the reproducing kernels of these spaces and/or in terms of a generalized Fourier transform that is obtained from the solution of an associated inverse spectral problem. Subsequently, the projection of the past onto the future and the future onto the past is interpreted in terms of the range of appropriately defined Hankel operators and their adjoints, and, in the last chapter, assorted computations are carried out for rational spectral densities. The underlying mathematics needed to tackle this class of problems is developed in careful detail, but, to ease the reading, an attempt is made to avoid excessive generality. En route a number of results that, to the best of our knowledge, were only known for p = 1 are generalized to the case p> 1.
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| Subject
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Multivariate analysis.
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Prediction theory.
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MATHEMATICS-- Applied.
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MATHEMATICS-- Probability Statistics-- General.
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Multivariate analysis.
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Prediction theory.
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Mathematics
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Operator Theory
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Probability Theory and Stochastic Processes
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| Dewey Classification
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519.5/35
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| LC Classification
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QA278
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| Added Entry
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Dym, H., (Harry),1938-
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