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

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dc.contributor.author Beneteau, Catherine
dc.contributor.author Condori, Alberto
dc.contributor.author Liaw, Constanze
dc.contributor.author Seco Forsnacke, Daniel
dc.contributor.author Sola, Alan A.
dc.date.accessioned 2021-06-07T10:52:59Z
dc.date.available 2021-06-07T10:52:59Z
dc.date.issued 2015-07
dc.identifier.bibliographicCitation First published in Pacific Journal of Mathematics in Vol. 276 (2015), No. 1, published by Mathematical Sciences Publishers
dc.identifier.issn 0030-8730
dc.identifier.uri http://hdl.handle.net/10016/32841
dc.description.abstract We 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.
dc.description.sponsorship Liaw 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.
dc.format.extent 24
dc.language.iso eng
dc.publisher Mathematical Sciences Publishers (MSP)
dc.rights © 2015 Mathematical Sciences Publishers
dc.subject.other Cyclicity
dc.subject.other Dirichlet-type spaces
dc.subject.other Optimal approximation
dc.subject.other Norm restrictions
dc.title Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk
dc.type article
dc.subject.eciencia Matemáticas
dc.identifier.doi https://doi.org/10.2140/pjm.2015.276.35
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2011-24606
dc.type.version publishedVersion
dc.identifier.publicationfirstpage 35
dc.identifier.publicationissue 1
dc.identifier.publicationlastpage 58
dc.identifier.publicationtitle Pacific Journal of Mathematics
dc.identifier.publicationvolume 276
dc.identifier.uxxi AR/0000026422
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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