Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk
| Autor: | Alan Sola, Constanze Liaw, Alberto A. Condori, Daniel Seco, Catherine Bénéteau |
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| Přispěvatelé: | Ministerio de Economía y Competitividad (España) |
| Rok vydání: | 2015 |
| Předmět: |
Norm restrictions
Sequence Pure mathematics Dirichlet-type spaces Mathematics - Complex Variables Matemáticas General Mathematics Zero (complex analysis) Hilbert space Type (model theory) 16. Peace & justice Dirichlet space Functional Analysis (math.FA) Mathematics - Functional Analysis Primary: 32A37 Secondary: 32A36 47A16 symbols.namesake FOS: Mathematics symbols Cyclicity Optimal approximation Complex Variables (math.CV) Unit (ring theory) Mathematics Variable (mathematics) Analytic function |
| Zdroj: | e-Archivo: Repositorio Institucional de la Universidad Carlos III de Madrid Universidad Carlos III de Madrid (UC3M) e-Archivo. Repositorio Institucional de la Universidad Carlos III de Madrid instname |
| ISSN: | 0030-8730 |
| DOI: | 10.2140/pjm.2015.276.35 |
| Popis: | We study Dirichlet-type spaces $\mathfrak{D}_{\alpha}$ of analytic functions in the unit bidisk and their cyclic elements. These are the functions $f$ for which there exists a sequence $(p_n)_{n=1}^{\infty}$ of polynomials in two variables such that $\|p_nf-1\|_{\alpha}\to 0$ as $n\to \infty$. We obtain a number of conditions that imply cyclicity, and obtain sharp estimates on the best possible rate of decay of the norms $\|p_nf-1\|_{\alpha}$, in terms of the degree of $p_n$, for certain classes of functions using results concerning Hilbert spaces of functions of one complex variable and comparisons between norms in one and two variables. We give examples of polynomials with no zeros on the bidisk that are not cyclic in $\mathfrak{D}_{\alpha}$ for $\alpha>1/2$ (including the Dirichlet space); this is in contrast with the one-variable case where all non-vanishing polynomials are cyclic in Dirichlet-type spaces that are not algebras ($\alpha\le 1$). Further, we point out the necessity of a capacity zero condition on zero sets (in an appropriate sense) for cyclicity in the setting of the bidisk, and conclude by stating some open problems. Comment: 20 pages |
| Databáze: | OpenAIRE |
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