Skip to main content

Advertisement

SpringerLink
Log in

On Higher-Order Szegő Theorems with a Single Critical Point of Arbitrary Order

  • Published:
Constructive Approximation Aims and scope

Abstract

We prove the following higher-order Szegő theorem: If a measure on the unit circle has absolutely continuous part \(w(\theta )\) and Verblunsky coefficients \(\alpha \) with square-summable variation, then for any positive integer m,

is finite if and only if \(\alpha \in \ell ^{2m+2}\). This is the first known equivalence result of this kind in the regime of very slow decay, i.e., with \(\ell ^p\) conditions with arbitrarily large p. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203(2), 341–347 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Denisov, S.A.: On a conjecture by Y. Last. J. Approx. Theory 158(2), 194–213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Golinskii, L., Zlatoš, A.: Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems. Constr. Approx. 26(3), 361–382 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaluzhny, U., Shamis, M.: Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square-summable variation. Constr. Approx. 35(1), 89–105 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math. (2) 158(1), 253–321 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Killip, R., Simon, B.: Sum rules and spectral measures of Schrödinger operators with \(L^2\) potentials. Ann. of Math. (2) 170(2), 739–782 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kreimer, Y., Last, Y., Simon, B.: Monotone Jacobi parameters and non-Szegő weights. J. Approx. Theory 157, 144–171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kupin, S.: On a spectral property of Jacobi matrices. Proc. Am. Math. Soc. 132(5), 1377–1383 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kupin, S.: Spectral properties of Jacobi matrices and sum rules of special form. J. Funct. Anal. 227(1), 1–29 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Laptev, A., Naboko, S., Safronov, O.: On new relations between spectral properties of Jacobi matrices and their coefficients. Commun. Math. Phys. 241(1), 91–110 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lukic, M.: Square-summable variation and absolutely continuous spectrum, J. Spectr. Theory, to appear. arXiv:1303.4161

  13. Lukic, M.: Orthogonal polynomials with recursion coefficients of generalized bounded variation. Commun. Math. Phys. 306(2), 485–509 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lukic, M.: On a conjecture for higher-order Szegő theorems. Constr. Approx. 38, 161–169 (2013). arXiv:1210.6953

    Article  MathSciNet  MATH  Google Scholar 

  15. Molchanov, S., Novitskii, M., Vainberg, B.: First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator. Commun. Math. Phys. 216(1), 195–213 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nazarov, F., Peherstorfer, F., Volberg, A., Yuditskii, P.: On generalized sum rules for Jacobi matrices. Int. Math. Res. Notes 3, 155–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Peherstorfer, F., Steinbauer, R.: Orthogonal polynomials on the circumference and arcs of the circumference. J. Approx. Theory 102(1), 96–119 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rakhmanov, E.A.: On the asymptotics of the ratio of orthogonal polynomials, ii. Math. USSR 46, 105–117 (1983)

    Article  MATH  Google Scholar 

  19. Simon, B.: Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, (2005), Classical theory

  20. Simon, B.: Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, (2005), Spectral theory

  21. Simon, B., Zlatoš, A.: Higher-order Szegő theorems with two singular points. J. Approx. Theory 134(1), 114–129 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Szegő, G.: Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 6(3–4), 167–202 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  23. Szegő, G.: Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 9(3–4), 167–190 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  24. Verblunsky, S.: On positive harmonic functions: a contribution to the algebra of Fourier series. Proc. Lond. Math. Soc. S2–38(1), 125–157 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  25. Verblunsky, S.: On positive harmonic functions (second paper). Proc. Lond. Math. Soc. 40, 290–320 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  26. Weidmann, J.: Zur Spektraltheorie von Sturm-Liouville-Operatoren. Math. Z. 98, 268–302 (1967)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rice University, 6100 Main Street, Mathematics MS 136, Houston, TX, 77005, USA

    Milivoje Lukic

Authors
  1. Milivoje Lukic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Milivoje Lukic.

Additional information

Communicated by Vilmos Totik.

The author was partially supported by NSF Grant DMS-1301582.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lukic, M. On Higher-Order Szegő Theorems with a Single Critical Point of Arbitrary Order. Constr Approx 44, 283–296 (2016). https://doi.org/10.1007/s00365-015-9320-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-015-9320-4

Keywords

Mathematics Subject Classification

Search

Navigation