- JOURNAL OF APPROXIMATION THEORY Journal
- September 2014
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- 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.
- measurable diagonalization; positive definite matrices; asymptotic; sobolev orthogonal polynomials; extremal polynomials; weighted sobolev spaces