Cyclicity in Dirichlet-type spaces and extremal polynomials

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.CB, DS, and AS would like to thank the Institut Mittag-Leffler and the AXA Research Fund for support while working on this project. CL is partially supported by the NSF grant DMS-1261687. DS is supported by the MEC/MICINN grant MTM-2008-00145. AS acknowledges support from the EPSRC under grant EP/103372X/1

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

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.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

Cyclic polynomials in two variables

We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves.The second author was supported by NSF grant DMS-1363239. The third author was supported by NCN grant 2011/03/B/ST1/04758. The fourth author was partially supported by NSF grant DMS-1261687. The fifth author was supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013 and MEC/MICINN Project MTM2011-24606. The sixth author acknowledges support from the EPSRC under grant EP/103372X/1

Remarks on Inner Functions and Optimal Approximants

We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to 1/f , where f is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials P minimizing Dirichlet‐type norms ∥pf−1∥α for a given function f . For α ∈ [0,1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non‐vanishing in the closed unit disk. For negative α , the weighted Bergman space case, the extremal polynomials are non‐vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how dist Dα (1, f · Pn) , where Pn is the space of polynomials of degree at most n , can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question.This work was supported by NSF under the grant DMS1500675. DS was supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013 and MEC Projects MTM2014-51824-P and MTM2011-24606