Overview
- 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
Part of the book series: Operator Theory: Advances and Applications (OT, volume 266)
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Table of contents (11 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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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Bibliographic Information
Book Title: Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems
Authors: Damir Z. Arov, Harry Dym
Series Title: Operator Theory: Advances and Applications
DOI: https://doi.org/10.1007/978-3-319-70262-9
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG, part of Springer Nature 2018
Hardcover ISBN: 978-3-319-70261-2Published: 09 June 2018
Softcover ISBN: 978-3-030-09944-2Published: 25 December 2018
eBook ISBN: 978-3-319-70262-9Published: 30 May 2018
Series ISSN: 0255-0156
Series E-ISSN: 2296-4878
Edition Number: 1
Number of Pages: XIV, 405
Topics: Operator Theory, Probability Theory and Stochastic Processes