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Abstract Band Method Via Factorization, Positive and Band Extensions of Multivariable Almost Periodic Matrix Functions and Spectral Estimation (Paperback)
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Abstract Band Method Via Factorization, Positive and Band Extensions of Multivariable Almost Periodic Matrix Functions and Spectral Estimation (Paperback)
Series: Memoirs of the American Mathematical Society
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New versions are developed of an abstract scheme, which are
designed to provide a framework for solving a variety of extension
problems. The abstract scheme is commonly known as the band method.
The main feature of the new versions is that they express directly
the conditions for existence of positive band extensions in terms
of abstract factorizations (with certain additional properties).
The results allow us to prove, among other things, that the band
extension is continuous in an appropriate sense. Using the new
versions of the abstract band method, we solve the positive
extension problem for almost periodic matrix functions of several
real variables with Fourier coefficients indexed in a given
additive subgroup of the space of variables.This generality allows
us to treat simultaneously many particular cases, for example the
case of functions periodic in some variables and almost periodic in
others. Necessary and sufficient conditions are given for the
existence of positive extensions in terms of Toeplitz operators on
Besikovitch spaces. Furthermore, when a solution exists a special
extension (the band extension) is constructed which enjoys a
maximum entropy property.A linear fractional parameterization of
the set of all extensions is also provided. We interpret the
obtained results (in the periodic case) in terms of existence of a
multivariate autoregressive moving averages (ARMA) process with
given autocorrelation coefficients, and identify its maximal
prediction error. Another application concerns the solution of the
positive extension problem in the context of Wiener algebra of
infinite operator matrices. It includes the identification of the
maximum entropy extension and a description of all positive
extensions via a linear fractional formula. In the periodic case it
solves a linear estimation problem for cyclostationary stochastic
processes.
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