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Measurable diagonalization of positive definite matrices

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2014-09-01
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Elsevier
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Abstract
In this paper we show that any positive definite matrix V with measurable entries can be written as V = U Lambda U*, where the matrix Lambda is diagonal, the matrix U is unitary, and the entries of U and Lambda are measurable functions (U* denotes the transpose conjugate of U). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.
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Measurable diagonalization, Positive definite matrices, Asymptotic, Sobolev orthogonal polynomials, Extremal polynomials, Weighted Sobolev spaces
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Journal of Approximation Theory, 2014, v. 185, pp. 91-97.