On Factorization of Trigonometric Polynomials |
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Authors: | Email author" target="_blank">Michael?A?DritschelEmail author |
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Institution: | (1) Department of Mathematics, School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK |
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Abstract: | We give a new proof of the operator version of the Fejér-Riesz
Theorem using only ideas from elementary operator theory. As an outcome,
an algorithm for computing the outer polynomials that appear in the Fejér-Riesz
factorization is obtained. The extremal case, where the outer factorization
is also *-outer, is examined in greater detail. The connection to Agler s
model theory for families of operators is considered, and a set of families lying
between the numerical radius contractions and ordinary contractions is
introduced. The methods are also applied to the factorization of multivariate
operator-valued trigonometric polynomials, where it is shown that the factorable
polynomials are dense, and in particular, strictly positive polynomials
are factorable. These results are used to give results about factorization of
operator valued polynomials over
, in terms of rational functions
with fixed denominators. |
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Keywords: | Primary: 47A68 Secondary: 47A65 42A05 30C10 15A23 |
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