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}\in L^p(\C)$, $p\geq 1 $, 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 $\Delta\phi$. They generalise previous results by Seip for the case $\phi(z)=|z|^2$, Berndtsson and Ortega-Cerdà and Ortega-Cerdà and Seip for the case when $\Delta\phi$ is bounded above and below, and Lyubarskiĭ and Seip for 1-homogeneous weights of the form $\phi(z)=|z|h(\arg z)$, where h is a trigonometrically strictly convex function.     

Flip-flop operation in opto-thermal bistable devices with localized absorption

We consider nonlinear interferometric devices based on opto-thermal bistability with localized absorption and show how the introduction of composite structures within the interferometer makes feasible its setting and resetting between stable outputs by applying positive light pulses on different parts of the bistable device. Experimental demonstration of all-optical flip-flop operation is reported for two interferometric configurations, one of them being extremely simple and well suited for parallel processing.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Zero sets of holomorphic functions in the unit ball with slow growth

We study the zero-varieties of holomorphic functions in the unit ball satisfying the growth condition log |f(z)|≤c _fλ(|z|), where λ:(0,1)→ℝ⁺ is a positive increasing function. We obtain some sufficient conditions on an analytic variety to be defined by such a function. Some results for the particular case λ(r)=log(e/(1−r)), corresponding to the classA ^−∞, generalize those of B. Korenblum in one variable. Both authors supported by DGICYT grant PB92-0804-C02-02.     

Traces of the Nevanlinna Class on Discrete Sequences

We show that a discrete sequence \(\Lambda \) of the unit disk is the union of n interpolating sequences for the Nevanlinna class \(\mathcal {N}\) if and only if the trace of \(\mathcal {N}\) on \(\Lambda \) coincides with the space of functions on \(\Lambda \) for which the divided differences of order \(n-1\) are uniformly controlled by a positive harmonic function.     

Density conditions for interpolation inA −∞

We give a necessary and sufficient density condition for a sequence {a _k} _k in the unit disk ofC to be interpolating for the classA ^−∞ of holomorphic functions with polynomial growth. The condition goes along the lines of those found by Seip for the analogous problem in Bergman spaces of the disk and by Berenstein and Li for some weighted spaces of entire functions. The result complements an earlier characterization given by Bruna and Pascuas. Partially supported by DGICYT grant PB95-0956-C02-02 and CIRIT grant 1998 SGR 00052. Also supported by a program of the Comunitat de Treball dels Pirineus.