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Cyclicity in Dirichlet-type spaces and extremal polynomials

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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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Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Florida, Tampa, FL, 33620, USA

    Catherine Bénéteau

  2. Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL, 33965-6565, USA

    Alberto A. Condori

  3. Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USA

    Constanze Liaw

  4. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Daniel Seco

  5. Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK

    Alan A. Sola

Authors
  1. Catherine Bénéteau
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  2. Alberto A. Condori
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  3. Constanze Liaw
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  4. Daniel Seco
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  5. Alan A. Sola
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Corresponding author

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

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