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Cyclicity in Dirichlet-type spaces and extremal polynomials

Abstract

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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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). https://doi.org/10.1007/s11854-015-0017-1

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  • DOI: https://doi.org/10.1007/s11854-015-0017-1

Keywords

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