Cyclicity in Dirichlet-type spaces and extremal polynomials

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Publication date
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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