Interpolation schemes in weighted Bergman spaces

The Michigan Mathematical Journal, 1996

Interpolating sequences for weighted Bergman spaces B p α , 0 < p ≤ ∞, α ≥ −1/p are studied. We show that the natural inclusions between B p α for various p and α are also verified by the corresponding spaces of interpolating sequences. We also give conditions (necessary or sufficient) for the B p α -interpolating sequences. These are similar to the known conditions for the spaces H p and A −α , which in our notation correspond respectively to the particular cases α = −1/p and p = ∞.

Most characterizations of interpolating sequences for Bergman spaces include the condition that the sequence be uniformly discrete in the hyperbolic metric. We show that if the notion of interpolation is suitably generalized, two of these characterizations remain valid without that condition. The general interpolation we consider here includes the usual simple interpolation and multiple interpolation as special cases.

Journal of Functional Analysis

We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case p = 2 and the uniform case p = +∞.

Comptes Rendus Mathematique, 2003

We prove that the complex interpolation space [A p 0 ν , A p 1 ν ] θ , 0 < θ < 1, between two weighted Bergman spaces A p 0 ν and A p 1 ν on the tube in C n , n 3, over an irreducible symmetric cone of R n is the weighted Bergman space A p ν with 1/p = (1 − θ)/p 0 + θ/p 1. Here, ν > n/r − 1 and 1 p 0 < p 1 < 2 + ν/(n/r − 1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D.

Given a finite set \sigma of the unit disc \mathbb{D}=\{z\in\mathbb{C}:,\,\vert z\vert&lt;1\} and a holomorphic function f in \mathbb{D} which belongs to a class X, we are looking for a function g in another class Y (smaller than X) which minimizes the norm \left\Vert g\right\Vert _{Y} among all functions g such that g_{\vert\sigma}=f_{\vert\sigma}. For Y=H^{\infty}, X=l_{a}^{p}(w_{k})=\,\left\{ f={\displaystyle \sum_{k\geq0}\hat{f}(k)z^{k}:\,\Vert f\Vert^{p}=\,{\displaystyle \sum_{k\geq0}\vert\hat{f}(k)\vert^{p}w_{k}^{p}&lt;\infty}}\right\} , with a weight w satisfying w_{k}&gt;0 for every k\geq0 and \overline{lim}_{k}(1/w_{k}^{1/k})=\,1, and for the corresponding interpolation constant c\left(\sigma,\, X,\, H^{\infty}\right), we show that if p=2, c\left(\sigma,\, X,\, H^{\infty}\right)\leq a\varphi_{X}\left(1-\frac{1-r}{n}\right) where n=\#\sigma, r=max_{\lambda\in\sigma}\left|\lambda\right| and where \varphi_{X}(t) stands for the norm of the evaluation functional f\mapsto f(t) on...

Rocky Mountain Journal of Mathematics, 2007

Let ρ denote the pseudohyperbolic metric in the unit disk D in the complex plane. We give examples of analytic functions g satisfying the condition |g(z)| ρ(z, Γ)(1 − |z|) −α , z ∈ D, in the case when Γ are A p zero sets considered by Horowitz and Luecking. This helps to solve directly interpolating and sampling problems for these sequences.

2020

We introduce a la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called disc polynomials that form an orthogonal basis of the whole weighted Hilbert space.

2003

The author showed that a sequence Z in the unit disk is a zero sequence for the Bergman space A p if and only if the weighted space L p (e pkZ dA) contains a non-zero (equivalently, zero-free) analytic function, where k Z (z) = a∈Z (1 − |a| 2) 2 |1 −āz| 2 |z| 2 2. Here we show that Z is an interpolating sequence for A p if and only if it is separated in the hyperbolic metric and the∂-equation (1 − |z| 2)∂u = f has a solution u satisfying u p,Z ≤ C f p,Z for every f ∈ L p (e pkZ dA), where f p,Z denotes the L p (e pkZ) norm. This holds for all finite p ≥ 1, but also for p < 1 if L p (e pkZ dA) is suitably modified. In addition, we show how this relates to a recent criterion for weighted∂ estimates by J. Ortega-Cerdà, and to K. Seip's criterion for interpolation.

Contemporary Mathematics, 2006

Let ρ : (0, 1] → R + be a weight function and let X be a complex Banach space. We denote by A 1,ρ (D) the space of analytic functions in the disc D such that D |f (z)|ρ(1 − |z|)dA(z) < ∞ and by Bloch ρ (X) the space of analytic functions in the disc D with values in X such that sup |z|<1 1−|z| ρ(1−|z|) F (z) < ∞. We prove that, under certain assumptions on the weight, the space of bounded operators L(A 1,ρ (D), X) is isomorphic to Bloch ρ (X) and some applications of this result are presented. Several properties of generalized vector-valued Bloch functions are also considered.

Advances in Operator Theory, 2020

The generalized weighted Bergman space HðB d ; kÞ is defined as a reproducing kernel Hilbert space of holomorphic functions on the open unit ball B d C d for all k [ 0. When k [ d, it is identical to the weighted Bergman space HL 2 ðB d ; l k Þ. We prove that the dual space HðB d ; aÞ Ã can be identified with another generalized weighted Bergman space HðB d ; bÞ under the pairing hf ; gi c ¼ R B d A k f ðzÞB k gðzÞ dl cþ2n ðzÞ; for f 2 HðB d ; aÞ; g 2 HðB d ; bÞ; where n ¼ d 2 AE Ç ; c ¼ aþb 2 and A k ; B k are operators related to the number operator N ¼ P d i¼1 z i o oz i :