Monotone Jacobi parameters and non-Szego{double acute} weights

Yury Kreimer; Yoram Last; Barry Simon

doi:10.1016/j.jat.2008.04.020

Monotone Jacobi parameters and non-Szego{double acute} weights

Yury Kreimer, Yoram Last, Barry Simon^*

^*Corresponding author for this work

Einstein Institute of Mathematics

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

5 Scopus citations

Abstract

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n ≡ 1, b_n = - C n^{- β} (0 < β < frac(2, 3)), one has d μ (x) = w (x) d x on (- 2, 2), and near x = 2, w (x) = e^{- 2 Q (x)} where Q (x) = β^{- 1} C^{frac(1, β)} frac(Γ (frac(3, 2)) Γ (frac(1, β) - frac(1, 2)) (2 - x)^{frac(1, 2) - frac(1, β)}, Γ (frac(1, β) + 1)) (1 + O ((2 - x))) .

Original language	English
Pages (from-to)	144-171
Number of pages	28
Journal	Journal of Approximation Theory
Volume	157
Issue number	2
DOIs	https://doi.org/10.1016/j.jat.2008.04.020
State	Published - Apr 2009

Bibliographical note

Funding Information:
The first author was supported in part by The Israel Science Foundation (grant no. 1169/06). The third author was supported in part by NSF grant DMS-0140592. The second and the third authors’ research was supported in part by Grant No. 2002068 and No. 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. It is a pleasure to thank Fritz Gesztesy, Uri Kaluzhny, and Doron Lubinsky for useful discussions. The authors would also like to thank Mira Shamis and the referees for pointing out several corrections to the original manuscript. B.S. would like to thank Ehud de Shalit for the hospitality of the Einstein Institute of Mathematics at the Hebrew University where some of this work was done. Y.L. would like to thank Matthias Flach for the hospitality of the Department of Mathematics at Caltech where some of this work was done.

Keywords

Orthogonal polynomials
Schrödinger operators
Spectral weights
Szego{double acute} condition

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10.1016/j.jat.2008.04.020

Cite this

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title = "Monotone Jacobi parameters and non-Szego{double acute} weights",

abstract = "We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for an ≡ 1, bn = - C n- β (0 < β < frac(2, 3)), one has d μ (x) = w (x) d x on (- 2, 2), and near x = 2, w (x) = e- 2 Q (x) where Q (x) = β- 1 Cfrac(1, β) frac(Γ (frac(3, 2)) Γ (frac(1, β) - frac(1, 2)) (2 - x)frac(1, 2) - frac(1, β), Γ (frac(1, β) + 1)) (1 + O ((2 - x))) .",

keywords = "Orthogonal polynomials, Schr{\"o}dinger operators, Spectral weights, Szego{double acute} condition",

author = "Yury Kreimer and Yoram Last and Barry Simon",

note = "Funding Information: The first author was supported in part by The Israel Science Foundation (grant no. 1169/06). The third author was supported in part by NSF grant DMS-0140592. The second and the third authors{\textquoteright} research was supported in part by Grant No. 2002068 and No. 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. It is a pleasure to thank Fritz Gesztesy, Uri Kaluzhny, and Doron Lubinsky for useful discussions. The authors would also like to thank Mira Shamis and the referees for pointing out several corrections to the original manuscript. B.S. would like to thank Ehud de Shalit for the hospitality of the Einstein Institute of Mathematics at the Hebrew University where some of this work was done. Y.L. would like to thank Matthias Flach for the hospitality of the Department of Mathematics at Caltech where some of this work was done. ",

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AU - Last, Yoram

AU - Simon, Barry

N1 - Funding Information: The first author was supported in part by The Israel Science Foundation (grant no. 1169/06). The third author was supported in part by NSF grant DMS-0140592. The second and the third authors’ research was supported in part by Grant No. 2002068 and No. 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. It is a pleasure to thank Fritz Gesztesy, Uri Kaluzhny, and Doron Lubinsky for useful discussions. The authors would also like to thank Mira Shamis and the referees for pointing out several corrections to the original manuscript. B.S. would like to thank Ehud de Shalit for the hospitality of the Einstein Institute of Mathematics at the Hebrew University where some of this work was done. Y.L. would like to thank Matthias Flach for the hospitality of the Department of Mathematics at Caltech where some of this work was done.

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N2 - We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for an ≡ 1, bn = - C n- β (0 < β < frac(2, 3)), one has d μ (x) = w (x) d x on (- 2, 2), and near x = 2, w (x) = e- 2 Q (x) where Q (x) = β- 1 Cfrac(1, β) frac(Γ (frac(3, 2)) Γ (frac(1, β) - frac(1, 2)) (2 - x)frac(1, 2) - frac(1, β), Γ (frac(1, β) + 1)) (1 + O ((2 - x))) .

AB - We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for an ≡ 1, bn = - C n- β (0 < β < frac(2, 3)), one has d μ (x) = w (x) d x on (- 2, 2), and near x = 2, w (x) = e- 2 Q (x) where Q (x) = β- 1 Cfrac(1, β) frac(Γ (frac(3, 2)) Γ (frac(1, β) - frac(1, 2)) (2 - x)frac(1, 2) - frac(1, β), Γ (frac(1, β) + 1)) (1 + O ((2 - x))) .

KW - Orthogonal polynomials

KW - Schrödinger operators

KW - Spectral weights

KW - Szego{double acute} condition

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Monotone Jacobi parameters and non-Szego{double acute} weights

Abstract

Bibliographical note

Keywords

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