Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band
The paper is concerned with Wiener algebra functions of two variables defined on the bitorus with an absolutely summable Fourier expansion. The problem of finding a positive definite matrix-valued function with certain prescribed coefficients is solved for the case when the coefficients are in a band bordered by two parallel lines that are of equal distance to the origin. A parametrization of such functions is obtained, and there exists one that maximizes the entropy, for which a formula is given. The study contains six sections. The first presents the main results. The irrational case is given in the section 2, using almost periodic function theory, and the rational case is given in section 3, reducing the problem to an operator-valued one-variable case. Some extension of the results are presented in section 4, while in section 5 a version of the extension problem for positive measures is given. In the last section, some of the results are interpreted in the context of spectral estimation of autoregressive moving average processes.
scientific article; zbMATH DE number 1275885
scientific article