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The Nevai Condition

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Abstract

We study Nevai’s condition that for orthogonal polynomials on the real line, \(K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}}\) , where K n is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.

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References

  1. Avila, A., Last, Y., Simon, B.: Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectra, preprint

  2. Boole, G.: On the comparison of transcendents, with certain applications to the theory of definite integrals. Philos. Trans. R. Soc. Lond. 147, 745–803 (1857)

    Article  Google Scholar 

  3. Breuer, J.: Spectral and dynamical properties of certain random Jacobi matrices with growing parameters. Trans. Am. Math. Soc. (to appear)

  4. Carmona, R.: One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types. J. Funct. Anal. 51, 229–258 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Christ, M., Kiselev, A., Last, Y.: Approximate eigenvectors and spectral theory. In: Differential Equations and Mathematical Physics. AMS/IP Stud. Adv. Math., vol. 16, pp. 85–96. American Mathematical Society, Providence (2000)

    Google Scholar 

  6. Christiansen, J.S., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, I. The isospectral torus. Constr. Approx. (to appear)

  7. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics. Springer, Berlin (1987); corrected and extended reprint (2008)

    MATH  Google Scholar 

  8. Damanik, D.: Strictly ergodic subshifts and associated operators. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th birthday. Proc. Sympos. Pure Math., vol. 76.2, pp. 505–538. American Mathematical Society, Providence (2007)

    Google Scholar 

  9. Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues. Commun. Math. Phys. 207, 687–696 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Damanik, D., Naboko, S.: Unbounded Jacobi matrices at critical coupling. J. Approx. Theory 145, 221–236 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Damanik, D., Killip, R., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, III. Alpha-continuity. Commun. Math. Phys. 212, 191–204 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Davis, B.: On the weak type (1,1) inequality for conjugate functions. Proc. Am. Math. Soc. 44, 307–311 (1974)

    MATH  Google Scholar 

  13. Denisov, S.A.: On a conjecture of Y. Last. J. Approx. Theory (to appear)

  14. Dombrowski, J.: Quasitriangular matrices. Proc. Am. Math. Soc. 69, 95–96 (1978)

    MATH  MathSciNet  Google Scholar 

  15. Freud, G.: Orthogonal Polynomials. Pergamon, Oxford (1971)

    Google Scholar 

  16. Jitomirskaya, S.: Singular spectral properties of a one-dimensional Schrödinger operator with almost periodic potential. In: Dynamical Systems and Statistical Mechanics, Moscow, 1991. Adv. Soviet Math., vol. 3, pp. 215–254. American Mathematical Society, Providence (1991)

    Google Scholar 

  17. Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra, I. Half-line operators. Acta Math. 183, 171–189 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kaluzhny, U., Last, Y.: Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation (in preparation)

  19. Kiselev, A., Last, Y., Simon, B.: Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Commun. Math. Phys. 194, 1–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Last, Y.: Destruction of absolutely continuous spectrum by perturbation potentials of bounded variation. Commun. Math. Phys. 274, 243–252 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Loomis, L.H.: A note on the Hilbert transform. Bull. Am. Math. Soc. 52, 1082–1086 (1946)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lubinksy, D.S.: A new approach to universality limits involving orthogonal polynomials. Ann. Math. (to appear)

  25. Lubinsky, D.S.: Universality limits in the bulk for arbitrary measures on compact sets. J. Anal. Math. 106, 373–394 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lubinsky, D.S., Nevai, P.: Sub-exponential growth of solutions of difference equations. J. Lond. Math. Soc. (2) 46, 149–160 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Maslov, V.P., Molchanov, S.A., Gordon, A.Ya.: Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture. Russ. J. Math. Phys. 1, 71–104 (1993)

    MATH  MathSciNet  Google Scholar 

  28. Máté, A., Nevai, P., Totik, V.: What is beyond Szegő’s theory of orthogonal polynomials? In: Rational Approximation and Interpolation, Tampa, FL, 1983. Lecture Notes in Math., vol. 1105, pp. 502–510. Springer, Berlin (1984)

    Chapter  Google Scholar 

  29. Máté, A., Nevai, P., Totik, V.: Extensions of Szegő’s theory of orthogonal polynomials. II. Constr. Approx. 3, 51–72 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  30. Máté, A., Nevai, P., Totik, V.: Extensions of Szegő’s theory of orthogonal polynomials. III. Constr. Approx. 3, 73–96 (1987)

    Article  MATH  Google Scholar 

  31. Máté, A., Nevai, P., Totik, V.: Szegő’s extremum problem on the unit circle. Ann. Math. 134, 433–453 (1991)

    Article  Google Scholar 

  32. Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 1–183 (1979)

    MathSciNet  Google Scholar 

  33. Nevai, P.: Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48, 3–167 (1986)

    Article  MATH  Google Scholar 

  34. Nevai, P., Totik, V., Zhang, J.: Orthogonal polynomials: their growth relative to their sums. J. Approx. Theory 67, 215–234 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Rakhmanov, E.A.: Steklov’s conjecture in the theory of orthogonal polynomials. Math. USSR-Sb. 36, 549–575 (1980). Russian original in: Mat. Sb. (N.S.) 108(150), 581–608, 640 (1979)

    Article  Google Scholar 

  36. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I. Functional Analysis. Academic Press, New York (1980)

    MATH  Google Scholar 

  37. Remling, C.: The absolutely continuous spectrum of Jacobi matrices, preprint

  38. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Am. Math. Soc. 124, 3361–3369 (1996)

    Article  Google Scholar 

  39. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Series, vol. 54.2. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  40. Simon, B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007)

    MATH  MathSciNet  Google Scholar 

  41. Simon, B.: Orthogonal polynomials with exponentially decaying recursion coefficients. In: Probability and Mathematical Physics. CRM Proc. Lecture Notes, vol. 42, pp. 453–463 (2007)

  42. Simon, B.: The Christoffel–Darboux kernel. In: Perspectives in PDE, Harmonic Analysis and Applications. Proc. Sympos. Pure Math., vol. 79, pp. 295–335. American Mathematical Society, Providence (2008)

    Google Scholar 

  43. Simon, B.: Two extensions of Lubinsky’s universality theorem. J. Anal. Math. 105, 345–362 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. Simon, B.: Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials (in preparation); to be published by Princeton University Press

  45. Stahl, H., Totik, V.: General orthogonal polynomials. In: Encyclopedia of Mathematics and Its Applications, vol. 43. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  46. Szegő, G.: Orthogonal Polynomials. Am. Math. Soc. Colloq. Publ., vol. 23. American Mathematical Society, Providence (1939); 3rd edn. (1967)

    Google Scholar 

  47. Szwarc, R.: A counterexample to subexponential growth of orthogonal polynomials. Constr. Approx. 11, 381–389 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  48. Szwarc, R.: Uniform subexponential growth of orthogonal polynomials. J. Approx. Theory 81, 296–302 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  49. Totik, V.: Asymptotics for Christoffel functions for general measures on the real line. J. Anal. Math. 81, 283–303 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  50. Totik, V.: Universality and fine zero spacing on general sets. Ark. Mat. (to appear)

  51. Zhang, J.: Relative growth of linear iterations and orthogonal polynomials on several intervals. Linear Algebra Appl. 186, 97–115 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  52. Zygmunt, M.J.: Some counterexamples to subexponential growth of orthogonal polynomials. Stud. Math. 116, 197–206 (1995)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    Jonathan Breuer & Barry Simon

  2. Institute of Mathematics, The Hebrew University, 91904, Jerusalem, Israel

    Yoram Last

Authors
  1. Jonathan Breuer
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  2. Yoram Last
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  3. Barry Simon
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Correspondence to Barry Simon.

Additional information

Communicated by Doron S. Lubinsky.

Research of the second author was supported in part by The Israel Science Foundation (Grant No. 1169/06) and Grant No. 2006483 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

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Breuer, J., Last, Y. & Simon, B. The Nevai Condition. Constr Approx 32, 221–254 (2010). https://doi.org/10.1007/s00365-009-9055-1

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