Synopses & Reviews
This book deals with the relation between the past and the future of a real, one-dimensional, stationary Gaussian process. Kolmogorov and Wiener showed how best to predict the future knowing the whole past. The more difficult problem, when only a finite segment of the past is known, was solved by M. G. Krein. A full treatment of this problem, and the prerequisites for dealing with it, occupies most of the book. The first three chapters are devoted to the necessary background in function theory, Hardy spaces and probability. Later chapters introduce the spectral theory of a weighted string developed by Krein and certain Hilbert spaces of entire functions introduced by L. de Branges. Various other connections between past and future are considered, such as mixing and Markovian character. The final chapter treats the problem of interpolation, when the whole process is known except for a gap and it is desired to predict what happens there.
Synopsis
This text offers background in function theory, Hardy functions, and probability as preparation for surveys of Gaussian processes, strings and spectral functions, and strings and spaces of integral functions. 1976 edition.
Synopsis
This text offers background in function theory, Hardy functions, and probability as preparation for surveys of Gaussian processes, strings and spectral functions, and strings and spaces of integral functions. 1976 edition.
Table of Contents
Authors' NotePrefaceIntroduction1. Background: Function Theory2. Background: Hardy Functions3. Background: Probability4. Past and Future5. Strings and Spectral Functions6. Strings and Spaces of Integral FunctionsReferencesIndex