Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class

B. Beckermann; A. Martínez-Finkelshtein; E. A. Rakhmanov; F. Wielonsky

doi:10.1063/1.1794842

Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class

B. Beckermann
, A. Martínez-Finkelshtein
, E. A. Rakhmanov
, F. Wielonsky

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We give an asymptotic upper bound as n→∞ for the entropy integral, E_n(w)=-∫p_n²(x)log(p _n²(x))w(x)dx, where p_n is the nth degree orthonormal polynomial with respect to a weight w(x) on [-1,1] which belongs to the Szego class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights w in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when w(x) belongs to the Szego class. In each case, we give conditions for these upper bounds to be attained.

Original language	English
Pages (from-to)	4239-4254
Number of pages	16
Journal	Journal of Mathematical Physics
Volume	45
Issue number	11
DOIs	https://doi.org/10.1063/1.1794842
State	Published - 1 Dec 2004
Externally published	Yes

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

Access to Document

10.1063/1.1794842

Cite this

@article{19f0eca445994aaeb4c3ba4dfefda34e,

title = "Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class",

abstract = "We give an asymptotic upper bound as n→∞ for the entropy integral, En(w)=-∫pn2(x)log(p n2(x))w(x)dx, where pn is the nth degree orthonormal polynomial with respect to a weight w(x) on [-1,1] which belongs to the Szego class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights w in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when w(x) belongs to the Szego class. In each case, we give conditions for these upper bounds to be attained.",

author = "B. Beckermann and A. Mart{\'i}nez-Finkelshtein and Rakhmanov, \{E. A.\} and F. Wielonsky",

year = "2004",

month = dec,

day = "1",

doi = "10.1063/1.1794842",

language = "English",

volume = "45",

pages = "4239--4254",

journal = "Journal of Mathematical Physics",

issn = "0022-2488",

publisher = "American Institute of Physics",

number = "11",

}

TY - JOUR

T1 - Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class

AU - Beckermann, B.

AU - Martínez-Finkelshtein, A.

AU - Rakhmanov, E. A.

AU - Wielonsky, F.

PY - 2004/12/1

Y1 - 2004/12/1

N2 - We give an asymptotic upper bound as n→∞ for the entropy integral, En(w)=-∫pn2(x)log(p n2(x))w(x)dx, where pn is the nth degree orthonormal polynomial with respect to a weight w(x) on [-1,1] which belongs to the Szego class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights w in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when w(x) belongs to the Szego class. In each case, we give conditions for these upper bounds to be attained.

AB - We give an asymptotic upper bound as n→∞ for the entropy integral, En(w)=-∫pn2(x)log(p n2(x))w(x)dx, where pn is the nth degree orthonormal polynomial with respect to a weight w(x) on [-1,1] which belongs to the Szego class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights w in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when w(x) belongs to the Szego class. In each case, we give conditions for these upper bounds to be attained.

UR - https://www.scopus.com/pages/publications/22544459932

U2 - 10.1063/1.1794842

DO - 10.1063/1.1794842

M3 - Article

AN - SCOPUS:22544459932

SN - 0022-2488

VL - 45

SP - 4239

EP - 4254

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 11

ER -

Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class

Abstract

ASJC Scopus subject areas

Access to Document

Fingerprint

Cite this