arXiv subjects

Pascal J. Thomas

Publications and source records attributed to Pascal J. Thomas.

63 records · Page 4Linked to original sources

arXiv2006

Sampling Sets for the Nevanlinna class

We propose a definition of sampling set for the Nevanlinna class in the disk, i.e. a subset of the disk such that the analogue of the norm of a function in the Nevanlinna class can be recovered only from its values on the subset. We show it is equivalent with the notion of determination set for the same class, that is, subsets such that any function in the class which is bounded on the subset must be bounded everywhere (by the same bound, in fact); and more restrictive than the condition of being a determination set for the class of differences of positive harmonic functions (which had been studied in particular by Hayman, Lyons, and Gardiner). We give sufficient conditions for sampling to hold, as well as necessary conditions (different but closely related), and show that in the case of certain (natural) regular sets, the necessary and sufficient conditions coincide and characterize the sampling property in a numerically precise way. In particular, we observe a remarkable agreement with the results of Joaquim Ortega-Cerdà and Kristian Seip about "champagne subdomains", the complement in the unit disk of a union of hyperbolic disks centered on a maximal hyperbolically separated subsequence, the radii of which decrease uniformly as they approach the unit circle (arXiv:math.CV/0305075). Our methods involve a careful analysis of the decrease of the modulus of a Blaschke product.

math.CV↗

arXiv2003

Equivalence of summatory conditions along sequences for bounded holomorphic functions

A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $ρ$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k ρ(1-|z_k|)|f(z_k)|$ converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained.

math.CV↗

arXiv2003

Harmonic and superharmonic majorants on the disk

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant.

math.CV↗

arXiv2002

Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.

math.CV↗

arXiv2002

Decrease of bounded holomorphic functions along discrete sets

We provide results of uniqueness for holomorphic functions in the Nevanlinna class bridging those previously obtained by Hayman and Lyubarskii-Seip. Namely, we propose certain classes of hyperbolically separated sequences in the disk, in terms of the rate of non-tangential accumulation to the boundary (the endpoints of this spectrum of classes being respectively the sequences with a non-tangential cluster set of positive measure, and the sequences violating the Blaschke condition); and for each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.

math.CV↗

arXiv2002

Pluricomplex Green and Lempert functions for equally weighted poles

For $Ω$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in Ω$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(Ω), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $Ω$ by $L_S (z) :=\inf \{\sum^N_{j=1}ν_j\log|ζ_j|: \exists ϕ\in \mathcal {O}(\mathbb D,Ω), ϕ(0)=z, ϕ(ζ_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $Ω$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.

math.CV↗

arXiv1997

Finite interpolation with minimum uniform norm in C^n

Given a finite sequence $a:={a_1, ..., a_N}$ in a domain $Ω\subset C^n$, and complex scalars $v:={v_1, ..., v_N}$, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough to contain the support of a measure whose hull contains a subset of the original $a$ large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which also contains this hull. An example is given to show that the inclusions can be strict.

math.CV↗

arXiv1996

Sampling sets for Hardy spaces of the disk

We propose two possible definitions for the notion of a sampling sequence (or set) for Hardy spaces of the disk. The first one is inspired by recent work of Bruna, Nicolau, and Øyma about interpolating sequences in the same spaces, and it yields sampling sets which do not depend on the value of $p$ and correspond to the result proved for bounded functions ($p=\infty$) by Brown, Shields and Zeller. The second notion, while formally closer to the one used for weighted Bergman spaces, is shown to lead to trivial situations only, but raises a possibly interesting problem.

math.CV↗

arXiv1995

Interpolating sequences for weighted Bergman spaces of the ball

Let $B_α^{p}$ be the space of $f$ holomorphic in the unit ball of $\Bbb C^n$ such that $(1-|z|^2)^αf(z) \in L^p$, where $0<p\leq\infty$, $α\geq -1/p$ (weighted Bergman space). In this paper we study the interpolating sequences for various $B_α^{p}$. The limiting cases $α=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and $A^{-α}$, the holomorphic functions with polynomial growth of order $α$, which have generated particular interest. In §1 we first collect some definitions and well-known facts about weighted Bergman spaces and then introduce the natural interpolation problem, along with some basic properties. In §2 we describe in terms of $α$ and $p$ the inclusions between $B_α^{p}$ spaces, and in §3 we show that most of these inclusions also hold for the corresponding spaces of interpolating sequences. §4 is devoted to sufficient conditions for a sequence to be $B_α^{p}$-interpolating, expressed in the same terms as the conditions given in previous works of Thomas for the Hardy spaces and Massaneda for $A^{-α}$. In particular we show, under some restrictions on $α$ and $p$, that finite unions of $B_α^{p}$-interpolating sequences coincide with finite unions of separated sequences. In his article in Inventiones, Seip implicitly gives a characterization of interpolating sequences for all weighted Bergman spaces in the disk. We spell out the details for the reader's convenience in an appendix (§5).

math.CV↗