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.
|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
Introduction.- Analytic preliminaries.- The de Branges spaces B(E) and H(A).- Three extension problems.- Spectral functions for ci problems.- Inverse spectral problems.- Generalizations.- Real and symmetric constraints.- Past and Future.- Conservative and passive systems.- Rational spectral densities.
"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)
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.
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.
Introduction.- Analytic preliminaries.- The de Branges spaces B(E) and H(A).- Three extension problems.- Spectral functions for ci problems.- Inverse spectral problems.- Generalizations.- Real and symmetric constraints.- Past and Future.- Conservative and passive systems.- Rational spectral densities.
"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)
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Inhaltsverzeichnis
Introduction.- Analytic preliminaries.- The de Branges spaces B(E) and H(A).- Three extension problems.- Spectral functions for ci problems.- Inverse spectral problems.- Generalizations.- Real and symmetric constraints.- Past and Future.- Conservative and passive systems.- Rational spectral densities.
Proposes solutions for prediction problemsDevelops the basics of Krein and de Branges theories in a new settingMakes the transition from one to several dimensions
