Operator Theory: Advances and Applications

Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems

Authors: Arov, Damir Z, Dym, Harry

  • 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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  • ISBN 978-3-319-70262-9
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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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Table of contents (11 chapters)

  • Introduction

    Arov, Damir Z. (et al.)

    Pages 1-26

  • Analytic Preliminaries

    Arov, Damir Z. (et al.)

    Pages 27-64

  • The de Branges Spaces $$\mathcal{B}\left(\mathfrak{E}\right)\;\mathrm{and}\;\mathcal{H}\left(A\right)$$

    Arov, Damir Z. (et al.)

    Pages 65-110

  • Three Extension Problems

    Arov, Damir Z. (et al.)

    Pages 111-152

  • Spectral Functions for Completely Indeterminate Problems

    Arov, Damir Z. (et al.)

    Pages 153-180

Buy this book

eBook $99.00
price for USA in USD (gross)
  • ISBN 978-3-319-70262-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $129.00
price for USA in USD
  • ISBN 978-3-319-70261-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems
Authors
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
Series ISSN
0255-0156
Edition Number
1
Number of Pages
XIV, 405
Topics