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.
Readership
Graduate students and research mathematicians.