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Cyclicity in Dirichlet-type spaces and extremal polynomials

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URI: https://hdl.handle.net/10016/32636
ISSN: 0021-7670
DOI: https://doi.org/10.1007/s11854-015-0017-1
UXXI: AR/0000026423
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Cyclicity_JAM_2015_ps.pdf (720.77 KB)
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2015-04
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Springer Nature
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Abstract
For functions f in Dirichlet-type spaces Dα, we study how to determine constructively optimal polynomials p n that minimize ∥pf−1∥α among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of ∥pnf−1∥α 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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Bergman space, Optimal norm, Open unit disk, Dirichlet space, Closed disk
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Bénéteau, C., Condori, A. A., Liaw, C., Seco, D. & Sola, A. A. (2015). Cyclicity in Dirichlet-type spaces and extremal polynomials. Journal d’Analyse Mathématique, 126(1), pp. 259–286.
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