Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems
Authors: Arov, Damir Z, Dym, Harry
Free Preview- Proposes solutions for prediction problems
- Develops the basics of Krein and de Branges theories in a new setting
- Makes the transition from one to several dimensions
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- About this book
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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.
- About the authors
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- Reviews
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“The book is a product of more than 20 years of collaborations between the two mathematicians belonging to Ukrainian and Isreali schools of functional analysis. They succeeded in writing a book on technically complicated subject that is easy to read. The book is strongly recommended to all researchers and graduate students in theoretical and applied probability and functional analysis … .” (Anatoliy Malarenko, Mathematical Reviews, February, 2019)
- Table of contents (11 chapters)
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Introduction
Pages 1-26
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Analytic Preliminaries
Pages 27-64
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The de Branges Spaces $$\mathcal{B}\left(\mathfrak{E}\right)\;\mathrm{and}\;\mathcal{H}\left(A\right)$$
Pages 65-110
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Three Extension Problems
Pages 111-152
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Spectral Functions for Completely Indeterminate Problems
Pages 153-180
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Table of contents (11 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems
- Authors
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- Damir Z Arov
- Harry Dym
- Series Title
- Operator Theory: Advances and Applications
- Series Volume
- 266
- Copyright
- 2018
- Publisher
- Birkhäuser Basel
- Copyright Holder
- Springer International Publishing AG, part of Springer Nature
- eBook ISBN
- 978-3-319-70262-9
- DOI
- 10.1007/978-3-319-70262-9
- Hardcover ISBN
- 978-3-319-70261-2
- Softcover ISBN
- 978-3-030-09944-2
- Series ISSN
- 0255-0156
- Edition Number
- 1
- Number of Pages
- XIV, 405
- Topics
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