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Cyclicity in Dirichlet-type spaces and extremal polynomials


For functions f in Dirichlet-type spaces \({D_\alpha }\), we study how to determine constructively optimal polynomials p n that minimize \({\left\| {pf - 1} \right\|_\alpha }\) among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of \({\left\| {{p_n}f - 1} \right\|_\alpha }\) as n approaches ∞, for certain classes of functions f. Finally, inspired by the Brown-Shields conjecture, we prove that certain logarithmic conditions on f imply cyclicity, and we study some computational phenomena pertaining to the zeros of optimal polynomials.

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  1. N. Arcozzi, R. Rochberg, E. T. Sawyer, and B. D. Wick, The Dirichlet space: a survey, New York J, Math. 17A (2011), 45–86.

    MATH  MathSciNet  Google Scholar 

  2. R. W. Barnard, J. Cima, and K. Pearce, Cesàro sum approximations of outer functions, Ann. Univ. Mariae Curie-SkŁodowska Sect. A 52 (1998), 1–7.

    MATH  MathSciNet  Google Scholar 

  3. R. W. Barnard, K. Pearce, and W. Wheeler, Zeros of Cesàro sum approximations, Complex Variables Theory Appl. 45 (2001), 327–348.

    MATH  MathSciNet  Article  Google Scholar 

  4. L. Brown, Invertible elements in the Dirichlet space, Canad. Math. Bull. 33 (1990), 419–422.

    MATH  MathSciNet  Article  Google Scholar 

  5. L. Brown and W. Cohn, Some examples of cyclic vectors in the Dirichlet space, Proc. Amer. Math. Soc. 95 (1985), 42–46.

    MATH  MathSciNet  Article  Google Scholar 

  6. L. Brown and A. L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), 269–304.

    MATH  MathSciNet  Article  Google Scholar 

  7. P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970.

    Google Scholar 

  8. P. L. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, Providence, RI., 2004.

    MATH  Book  Google Scholar 

  9. O. El-Fallah, K. Kellay, and T. Ransford, Cyclicity in the Dirichlet space, Ark. Mat. 44 (2006), 61–86.

    MATH  MathSciNet  Article  Google Scholar 

  10. O. El-Fallah, K. Kellay, and T. Ransford, On the Brown-Shields conjecture for cyclicity in the Dirichlet space, Adv. Math. 222 (2009), 2196–2214.

    MATH  MathSciNet  Article  Google Scholar 

  11. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000.

    MATH  Book  Google Scholar 

  12. H. Hedenmalm and A. Shields, Invariant subspaces in Banach spaces of analytic functions, Michigan Math. J. 37 (1990), 91–104.

    MATH  MathSciNet  Article  Google Scholar 

  13. S. Richter and C. Sundberg, Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory 28 (1992), 167–186.

    MATH  MathSciNet  Google Scholar 

  14. W. T. Ross, The classical Dirichlet space, in Recent Advances in Operator-Related Function Theory, Contemp. Math. 393 (2006), 171–197.

    Article  Google Scholar 

  15. A. R. Vargas, Zeros of sections of some power series, MSc. Thesis, Dalhousie University, 2012, arXiv:1208.5186v2[math.NT].

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Correspondence to Catherine Bénéteau.

Additional information

CB, DS, and AS would like to thank the Institut Mittag-Leffler and the AXA Research Fund for support while working on this project.

CL is partially supported by the NSF grant DMS-1261687.

DS is supported by the MEC/MICINN grant MTM-2008-00145.

AS acknowledges support from the EPSRC under grant EP/103372X/1.

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Bénéteau, C., Condori, A.A., Liaw, C. et al. Cyclicity in Dirichlet-type spaces and extremal polynomials. JAMA 126, 259–286 (2015).

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  • Bergman Space
  • Optimal Norm
  • Open Unit Disk
  • Dirichlet Space
  • Closed Disk