Christoffel Transforms and Hermitian Linear Functionals

Abstract

Abstract. In this contribution we are focused on some spectral transformations of Hermitian linear functionals. They are the analogues of the Christoffel transform for linear functionals, i. e. for Jacobi matrices which has been deeply studied in the past. We consider Hermitian linear functionals associated with a probability measure supported on the unit circle. In such a case we compare the Hessenberg matrices associated with such a probability measure and its Christoffel transform. In this way, almost unitary matrices appear. We obtain the deviation to the unit matrix both for principal submatrices and the complete matrices respectively. Mathematics Subject Classification (2000). Primary 42C05; Secondary 15A23.

Spectral transformations of measures supported on the unit circle and the Szegő transformation

“…The perturbation dσ = |z − α| 2 dσ , |z| = 1, α ∈ C, is the so-called canonical Christoffel transformation (see [10]). …”

Linear Spectral Transformations and Laurent Polynomials

“…In [4] and [16] we have studied the connection between the associated Hessenberg matrices using the QR factorization. The iteration of the canonical Christoffel transformation has been analyzed in [8], [11], and [14].…”

“…We recover the Christoffel transformation when |Re(α)| 1. This transformation was studied in [4], [16]. Now, we study the relation between the Hessenberg matrix associated with L R , which will be denoted by H Y , and the Hessenberg matrix associated with L.…”

“…2 dθ 2π , z = e iθ (see [16]). Is is well known that if {Φ n } n 0 denotes the sequence of monic polynomials orthogonal with respect to σ, then …”

Szegő and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations

“…The analogues on the unit circle to the canonical spectral transformations on the real line have been introduced by Marcellán and co-workers; see [3,5,6,9,10] and are known in the literature by Christoffel, Geronimus and Uvarov. Our goal in this section is to study the zeros of polynomials generated by these three perturbations for Szegő.…”