Singular Measures on the Unit Circle and Their Reflection Coefficients

In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (a n ). Among other cases, we study the derived set of the support of the measure when lim n |a n+1 /a n | = 1, obtaining a natural generalization of the known result for the López class lim n a n+1 /a n ∈ T, lim n |a n | ∈ (0, 1).On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials.Finally, we use the previous results to discuss the domain of convergence of rational approximants of Carathéodory functions, including the convergence on the unit circle.

show abstract

“…Therefore, in this case, {supp } = −L{a n+1 /a n } (see [13] for a similar argument using Hessenberg representations).…”

Section: Remark 36mentioning

confidence: 81%

“…, a n−1 , u) ⊕ C(a (n) ; u). If := C(a) − C(â) and := ( n ) n 0 , (13) and (14) lead to t (z) = z m (b n p u n (z) + d n q u n (z)), b n ∈ 2 n , d n ∈ 2⊥ n .…”

Section: Proofmentioning

confidence: 99%

Measures on the unit circle and unitary truncations of unitary operators

Cantero

Ituarte

Velázquez

2006

Journal of Approximation Theory

View full text Add to dashboard Cite

show abstract

“…In Section 7, we will use Theorems 3.2 and 3.3 to complement the analysis of Krein (which appeared in Akhiezer-Krein [3]) for bounded Jacobi matrices with finite essential spectrum, and of Golinskii [29] for OPUC with finite derived sets.…”

Section: The Essential Spectrummentioning

confidence: 99%

“…We have come to Golinskii's OPUC analog of Krein's theorem [29]. Again, it is illuminating to consider the case ℓ = 2.…”

Section: Additional Applicationsmentioning

confidence: 99%

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

Simon

2006

J. Anal. Math.

133

View full text Add to dashboard Cite

Abstract. We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein's results on orthogonal polynomials with finite essential spectra.

show abstract

“…almost everywhere with respect to Lebesgue measure m. It is clear from (12) that G 0 contains all singular measures with m(supp s)=0. For instance, the measures with finite derived set of support [14,Section 3] belong to G 0 .…”

Section: 1)mentioning

confidence: 99%

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

Golinskiĭ

Khrushchev

2002

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

The convergence in L 2 (T) of the even approximants of the Wall continued fractions is extended to the Cesàro-Nevai class CN, which is defined as the class of probability measures s with lim n Q . 1 n ; n − 1 k=0 |a k |=0, {a n } n \ 0 being the Geronimus parameters of s. We show that CN contains universal measures, that is, probability measures for which the sequence {|j n | 2 ds} n \ 0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the ''opposite'' Szegő class which consists of measures with ;.1/2 < . and describe it in terms of Hessenberg matrices. © 2002 Elsevier Science (USA)

show abstract

Singular Measures on the Unit Circle and Their Reflection Coefficients

Cited by 9 publications

References 23 publications

Measures on the unit circle and unitary truncations of unitary operators

Measures on the unit circle and unitary truncations of unitary operators

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

Contact Info

Product

Resources

About