# Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 3-35.

Contractive extension problems for matrix valued almost periodic functions of several variables**Authors**: Leiba Rodman (1), Ilya M. Spitkovsky (2), and Hugo J. Woerdeman (3)

**Author institution:**(1) Department of Mathematics, P.O. Box 8795, The College of William and Mary, Williamsburg, VA 23187--8795, USA

(2) Department of Mathematics, P.O. Box 8795, The College of William and Mary, Williamsburg, VA 23187--8795, USA

(3) Department of Mathematics, P.O. Box 8795, The College of William and Mary, Williamsburg, VA 23187--8795, USA

**Summary:**Problems of Nehari type are studied for matrix valued $k$-variable almost periodic Wiener functions: Find contractive $k$-variable almost periodic Wiener functions having prespecified Fourier coefficients with indices in a given halfspace of $ {{\bbb R}}^k$. We characterize the existence of a solution, give a construction of the solution set, and exhibit a particular solution that has a certain maximizing property. These results are used to obtain various distance formulas and multivariable almost periodic extensions of Sarason's theorem. In the periodic case, a generalization of Sarason's theorem is proved using a variation of the commutant lifting theorem. The main results are further applied to a model-matching problem for multivariable linear filters.

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