A Characterization of Dirichlet-inner Functions

We study a concept of inner function suited to Dirichlet-type spaces. We characterize Dirichlet-inner functions as those for which both the space and multiplier norms are equal to 1

A zk-invariant subspace without the wandering property

We study operators of multiplication by zk in Dirichlet-type spaces Dα. We establish the existence of k and α for which some zk-invariant subspaces of Dα do not satisfy the wandering property. As a consequence of the proof, any Dirichlet-type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by zk, for any k≥6

Smoothness of sets in Euclidean spaces

We study some properties of smooth sets in the sense defined by Hungerford. We prove a sharp form of Hungerford's theorem on the Hausdorff dimension of their boundaries on Euclidean spaces and show the invariance of the definition under a class of automorphisms of the ambient space.Both authors are supported partially by the grants MTM2008-00145 and 2009SGR420

Polynomial approach to cyclicity for weighted ℓpA

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called optimal polynomial approximants. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in ℓp(ω), for some weight ω. When ω={(k+1)α}k∈N, for a fixed α∈R, we derive a characterization of the cyclicity of polynomial functions and, when 1 < p < ∞, we obtain sharp rates of convergence of the optimal norms.We acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa Programme for Centers of Excellence in R&D" (SEV-2015-0554) and through grant MTM2016-77710-P. We are also grateful to Raymond Cheng and to an anonymous referee for helpful comments and careful reading

Schatten classes of generalized Hilbert operators

Let Dv denote the Dirichlet type space in the unit disc induced by a radial weight v for which vˆ(r)=∫1rv(s)ds satisfies the doubling property ∫1rv(s)ds≤C∫11+r2v(s)ds. In this paper, we characterize the Schatten classes Sp(Dv) of the generalized Hilbert operators Hg(f)(z)=∫10f(t)g′(tz)dt acting on Dv, where v satisfies certain Muckenhoupt type conditions. For p≥1, it is proved that Hg∈Sp(Dv) if and only if ∫10((1−r)∫π−π|g′(reiθ)|2dθ)p2dr1−r<∞

No Entire Inner Functions

We study generalized inner functions on a large family of ReproducingKernel Hilbert Spaces. We show that the only inner functions whichare entire are the normalized monomials

Cyclicity in Reproducing Kernel Hilbert Spaces of analytic functions

We introduce a large family of reproducing kernel Hilbert spaces H⊂Hol(D), which include the classical Dirichlet-type spaces Dα, by requiring normalized monomials to form a Riesz basis for H. Then, after precisely evaluating the nth optimal norm and the n-th approximant of f(z)=1−z, we completely characterize the cyclicity of functions in Hol(D¯¯¯¯) with respect to the forward shift.For this work, we were supported by grants from Labex CEMPI (ANR-11-LABX-0007-01), NSERC (100756), ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013, MEC/MICINN Project MTM2011-24606, and Generalitat de Catalunya 2009SGR420

Boundary Behavior of Optimal Polynomial Approximants

In this paper, we provide an efficient method for computing the Taylor coefficients of 1−pnf, where pn denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space H2ω of analytic functions over the unit disc D, and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces H2ω, the sequence {1−pnf}n∈N is uniformly bounded on the closed unit disc and, if f has no zeros inside D, the sequence {1−pnf} converges uniformly to 0 on compact subsets of the complement of the zeros of f in D, and we obtain precise estimates on the rate of convergence on compacta. We also obtain the precise constant in the rate of decay of the norm of 1−pnf in the previously unknown case of a function with a single zero of multiplicity greater than 1, when the weights are given by ωk=(k+1)α for α≤1.Myrto Manolaki thanks the Department of Mathematics and Statistics at the University of South Florida for support during work on this project. Daniel Seco acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centers of Excellence in R&D” (SEV-2015-0554) and through grant MTM2016-77710-P. The authors are grateful to the referees for their careful reading of the article and their useful comments

On the Wandering Property in Dirichlet spaces

We show that in a scale of weighted Dirichlet spaces Dα, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in Dα such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when B(z)=zk and |α|≤log(2)log(k+1), the chosen norm is the usual one in Dα

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.The authors would like to thank A. Sola for useful discussions, and the National Science Foundation for their support of the SEAM conference in 2016, where some of this work was carried out. Beneteau and Khavinson are grateful to the Centre de Recherches Mathematiques in Montreal for hosting them during the spring semester of 2016. Khavinson acknowledges the support of grant #513381 from the Simons Foundation. Liaw's work is supported by the National Science Foundation DMS-1802682. Seco acknowledges support from Ministerio de Economía y Competitividad Project MTM2014-51824-P and from Generalitat de Catalunya Project 2014SGR289