Interpolation sequences for the Bernstein algebra
We give a description, in analytic and geometric terms, of the interpolation sequences for the algebra of entire functions of exponential type which are bounded on the real line.
math.CV↗
arXiv subjects
Xavier Massaneda
Publications and source records attributed to Xavier Massaneda.
Traces of Hörmander algebras on discrete sequences
We show that a discrete sequence $Λ$ of the complex plane is the union of $n$ interpolating sequences for the Hörmander algebras $A_p$ if and only if the trace of $A_p$ on $Λ$ coincides with the space of functions on $Λ$ for which the divided differences of order $n-1$ are uniformly bounded. The analogous result holds in the unit disk for Korenblum-type algebras.
math.CV↗
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↗
Interpolation in $\hat{\E^\prime}(\R)$
We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.
math.CV↗
Interpolating sequences for the Nevanlinna and Smirnov classes
We give an analytic characterisation of the interpolating sequences for the Nevanlinna and Smirnov classes. From this we deduce a necessary and a sufficient geometric condition, both expressed in terms of a certain non-tangential maximal function associated to the sequence. Some examples show that the gap between the necessary and the sufficient condition cannot be covered. We also discuss the relationship between our results and the previous work of Naftalevic for the Nevanlinna class, and Yanagihara for the Smirnov class. Finally, we observe that the arguments used in the previous proofs show that interpolating sequences for ``big'' Hardy-Orlicz spaces are in general different from those for the scale included in the classical Hardy spaces.
math.CV↗
Interpolating and sampling sequences for entire functions
We characterise interpolating and sampling sequences for the spaces of entire functions f such that f e^{-phi} belongs to L^p(C), p>=1 (and some related weighted classes), where phi is a subharmonic weight whose Laplacian is a doubling measure. The results are expressed in terms of some densities adapted to the metric induced by the Laplacian of phi. They generalise previous results by Seip for the case phi(z)=|z|^2, and by Berndtsson & Ortega-Cerdà and Ortega-Cerdà & Seip for the case when the Laplacian of phi is bounded above and below.
math.CV↗
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↗