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Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorBeneteau, Catherine
dc.contributor.authorCondori, Alberto
dc.contributor.authorLiaw, Constanze
dc.contributor.authorSeco Forsnacke, Daniel
dc.contributor.authorSola, Alan A.
dc.contributor.funderMinisterio de Economía y Competitividad (España)es
dc.date.accessioned2021-06-07T10:52:59Z
dc.date.available2021-06-07T10:52:59Z
dc.date.issued2015-07
dc.description.abstractWe study Dirichlet-type spaces Dα of analytic functions in the unit bidisk and their cyclic elements. These are the functions f for which there exists asequence(pn)∞n=1 of polynomials in two variables such that ‖pnf−1‖α→0 as n→∞. We obtain a number of conditions that imply cyclicity, and obtain sharp estimates on the best possible rate of decay of the norms ‖pnf−1‖α,in terms of the degree of pn, 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 Dα for α >1/2 (including the Dirichlet space); this is in contrast with the one-variable case where all nonvanishing polynomials are cyclic in Dirichlet-type spaces that are not algebras (α≤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.en
dc.description.sponsorshipLiaw is partially supported by the NSF grant DMS-1261687. Seco is supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013, and by MEC/MICINN Project MTM2011-24606. Sola acknowledges support from the EPSRC under grant EP/103372X/1.en
dc.format.extent24
dc.identifier.bibliographicCitationFirst published in Pacific Journal of Mathematics in Vol. 276 (2015), No. 1, published by Mathematical Sciences Publishersen
dc.identifier.doihttps://doi.org/10.2140/pjm.2015.276.35
dc.identifier.issn0030-8730
dc.identifier.publicationfirstpage35
dc.identifier.publicationissue1
dc.identifier.publicationlastpage58
dc.identifier.publicationtitlePacific Journal of Mathematicsen
dc.identifier.publicationvolume276
dc.identifier.urihttps://hdl.handle.net/10016/32841
dc.identifier.uxxiAR/0000026422
dc.language.isoeng
dc.publisherMathematical Sciences Publishers (MSP)en
dc.relation.projectIDGobierno de España. MTM2011-24606es
dc.rights© 2015 Mathematical Sciences Publishersen
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.subject.otherCyclicityen
dc.subject.otherDirichlet-type spacesen
dc.subject.otherOptimal approximationen
dc.subject.otherNorm restrictionsen
dc.titleCyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisken
dc.typeresearch article*
dc.type.hasVersionVoR*
dspace.entity.typePublication
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