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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 andor 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 gt 1
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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 andor 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 gt 1
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