Measurable diagonalization of positive definite matrices

Yamilet Quintana ^[2] ; José Manuel Rodríguez García ^[1]
1. [1] Universidad Carlos III de Madrid
  
  Universidad Carlos III de Madrid
  
  Madrid, España
2. [2] Departamento de Matemáticas Puras y Aplicadas, Edificio Matemáticas y Sistemas (MYS), Apartado Postal: 89000, Caracas 1080 A, Universidad Simón Bolívar, Venezuela
Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 185, Nº 1, 2014, págs. 91-97
Idioma: inglés
DOI: 10.1016/j.jat.2014.06.003
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Resumen
- In this paper we show that any positive definite matrix V with measurable entries can be written as V=UɅU* , where the matrix is diagonal, the matrix U is unitary, and the entries of U and Ʌ 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.