Measurable diagonalization of positive definite matrices Articles uri icon

publication date

  • September 2014

start page

  • 91

end page

  • 97

volume

  • 185

international standard serial number (ISSN)

  • 0021-9045

electronic international standard serial number (EISSN)

  • 1096-0430

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.

keywords

  • measurable diagonalization; positive definite matrices; asymptotic; sobolev orthogonal polynomials; extremal polynomials; weighted sobolev spaces