Driss Aadi, Carlos Cruz, Andreas Hartmann, Karim Kellay
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 = +$\infty$.
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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.
We extend our work on nonseparated interpolating sequences, originally developed for Bergman spaces with weights of the form $(1 - |z|^2)^α$, to more general weights.
Duality and Interpolation of Bergman Spaces
本文研究p-Bergman空间的对偶空间及其为q-Bergman空间的本质条件，通过分析Banach偶的插值空间展开，并对比区域的对偶性、可积性与正则性性质，推进了Bergman空间理论研究。
We formulate the Bergman-type interpolation problem on finite open Riemann surfaces covered by the unit disk. Our version of the interpolation problem generalizes Bergman-type interpolation problems previously studied by Seip, Berntsson, Ortega Cerdà, and a number of other authors. We then prove necessary and sufficient conditions for interpolation, and also some sufficient conditions under even weaker hypotheses. The results extend work of Ortega Cerdà, who resolved the case in which the boundary of the surface is pure $1$-dimensional. Our version of the interpolation problem effectively chan…
We study the Bergman space interpolation problem of open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. We equip such a surface with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. We then establish necessary and sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces.
Let $A$ be a sequence of points of $\mathbb{B}^n$ the unit ball in $\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_α(\mathbb{B}^n). In the particular case of Hardy space $H^p (\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_α(\mathbb{B}^n)$
We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional…
Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-α}$, $0<α<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.
This paper extends the known characterization of interpolation and sampling sequences for Bergman spaces to the mixed-norm spaces. The Bergman spaces have conformal invariance properties not shared by the mixed-norm spaces. As a result, different techniques of proof were required.
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and …
The zero sets of the Bergman space $A^p_ω$ induced by either a radial weight $ω$ admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking's results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz's factorization result from 1977 for functions in $A^p_ω$. The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling mea…
We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.
We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. However, unlike previous work, we construct an infinite number of densities, naturally parameterized by $L^1_{loc} ([0,\…
There has been a great deal of work done in recent years on weighted Bergman spaces $\apa$ on the unit ball $\bn$ of $\cn$, where $0<p<\infty$ and $α>-1$. We extend this study in a very natural way to the case where $α$ is {\em any} real number and $0<p\le\infty$. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^2$, and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergm…
We discuss sampling constants for dominating sets in Bergman spaces. Our method is based on a Remez-type inequality by Andrievskii and Ruscheweyh. We also comment on extensions of the method to other spaces such as Fock and Paley-Wiener spaces.
It is well known that a function is in a Bergman space of the unit ball if and only if it satisfies some Hardy-type inequalities. We extend this fact to Bergman-Orlicz spaces. As applications, we obtain Gustavsson-Peetre interpolation of two Bergman-Orlicz spaces and we completely characterize symbols of bounded or compact Cesàro-type operators on Bergman-Orlicz spaces, extending known results for classical weighted Bergman spaces.
We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H $\infty$ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constan…
Stević-Sharma type operators between Bergman spaces induced by doubling weights
本文利用 Khinchin 不等式、Geršgorin 定理和原子分解，刻画倍增权加权 Bergman 空间上 Stević-Sharma 型及加权微分复合算子和的范数与本质范数，并获得插值定理。
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.
We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb{C}^d$.
We use induction and interpolation techniques to prove that a composition operator induced by a map $ϕ$ is bounded on the weighted Bergman space $\A^2_α(\mathbb{H})$ of the right half-plane if and only if $ϕ$ fixes $\infty$ non-tangentially, and has a finite angular derivative $λ$ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by $λ^{(2+α)/2}$.
On the Theory of Bergman Spaces on Homogeneous Siegel Domains
针对齐性Siegel域上混合范数Bergman空间现有两种研究思路下结果难对比的问题，本文梳理已有成果并在一种框架下补充了包含关系、对偶性等多方面结论，完善了相关理论。
We study multiple sampling, interpolation and uniqueness for the classical Fock space in the case of unbounded mul-tiplicities.
A sequence which is a finite union of interpolating sequences for $H^\infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f \mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also…
Let $0<p<\infty$, $β>-1$, and $Ω$ be a strongly pseudoconvex bounded domain with a smooth boundary in $\mathbb{C}^n$. We will study the interpolation problem for weighted Bergman spaces $A^p_β(Ω)$. In the case, $1\leq p<\infty$, and $β> \max \{n(2p-1)-1, n(2q-1)-1\}$, where $q$ is the conjugate exponent of $p$ (let $q=1$, for $p=1$), we show that a sequence in $\mathbb{B}_n$, the unit ball in $\mathbb{C}^n$, is interpolating for $A^p_β(\mathbb{B}_n)$ if and only if it is separated.
We study multiple sampling, interpolation and uniqueness for the classical Fock spaces in the case of unbounded multiplicities. We show that there are no sequences which are simultaneously sampling and interpolating when the multiplicities tend to infinity.
Interpolating sequences for weighted Bergman spaces of the ball
设$B_α^{p}$是$f$在$\Bbb C^n$的单位球上全纯的空间，使得$(1-|z|^2)^αf(z) \in L^p$，其中$0<p\leq\infty$，$α\geq -1/p$（加权Bergman空间）.本文研究了各种$B_α^{p}$的插值序列。极限情况$α=-1/p$和$p=\infty$分别是Hardy空间$H^p$和$A^{-α}$，全纯函数的多项式增长的顺序$α$，产生了特别的兴趣。在§1中，我们首先收集了一些关于加权Bergman空间的定义和众所周知的事实，然后介绍了自然插值问题，以及一些基本性质。在§2中，我们用$α$和$p$描述了$B_α^{p}$空间之间的包含，在§3中，我们证明了这些包含中的大多数对于相应的插值序列空间也成立。§4致力于序列是$B_α^{p}$ -内插的充分条件，用与Thomas对Hardy空间和Massaneda对$A^{-α}$的先前工作中给出的条件相同的术语表示。特别是，我们表明，$α$和$p$的一些限制下，$B_α^{p}$ -插值序列的有限工会符合分离序列的有限工会。在他的文章中Inventiones，塞普隐式地给出了一个特征的插值序列的所有加权伯格曼空间中的磁盘。为了读者的方便，我们在附录（第5节）中详细说明了这些细节。
Multiple sampling and interpolation in a space of polynomials
本文研究了多项式空间中具有多重性的采样与插值问题，发现其几何条件与Fock空间中无界多重性的条件一致，揭示了多项式空间在极限情况下的特性。
Sampling and interpolation for analytic selfmappings of the disc
本文研究了圆盘解析自映射的两个问题：给出了采样序列的刻画，并基于$n$点Schwarz-Pick引理导出了一个可视为经典Nevanlinna-Pick插值特例的插值问题。
We study those smooth complex hypersurfaces W in C^n having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in C^n such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric condi…
Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions
本文定义了基于 Dunkl 算子的 $λ$-解析函数，并研究了相关 Bergman 和 Hardy 空间的投影有界性、函数增长、稠密性、完备性、对偶空间及刻画与插值等问题。
Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy Spaces
本文研究无限维解析函数Hilbert空间中采样序列与多项式子空间Marcinkiewicz-Zygmund不等式的关系，重点比较单变量Hardy空间与Bergman空间中截然不同的行为。
We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequen…
Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in Harmonic Analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Möbius invariance for the tent spaces.
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.
Two different problems are considered here. First, a version of Schwarz-Pick Lemma for $n$ points leads to an interpolation problem for analytic functions from the disc into itself, which may be considered as a particular case of the classical Nevanlinna-Pick interpolation problem. Second, a characterization of sampling sequences for this class of functions is given.
Density theorems for sampling and interpolation in the Bargmann-Fock space
我们给出了一个完整的描述的采样和插值的Bargmann-Fock空间，密度概念的基础上的Beurling。粗略地说，一个离散集是一个采样集当且仅当它在平面的每一部分的密度都严格大于冯诺依曼格的密度，类似地，一个离散集是一个插值集当且仅当它在平面的每一部分的密度都严格小于冯诺依曼格的密度。
We denote by $\Hp$ the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane $σ>1/2$ is an interpolating sequence for $\Hp$ if and only if it is an interpolating sequence for the Hardy space $H^2$ of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane $σ>1/2$.
Dirichlet integral dual-access collocation-kernel space analytic interpolation for unit disks: DIDACKS I
提出一种利用内部复极点实现单位圆盘外部解析插值的双访问配点核空间技术，并扩展到实/复单位圆盘内外区域，同时考虑高阶极点和对数点源核，与Szego和Bergman核理论建立联系。
We solve an interpolation problem in $A^p_α$ involving specifying a set of (possibly not distinct) $n$ points, where the $k^{\textrm{th}}$ derivative at the $k^{\textrm{th}}$ point is up to a constant as large as possible for functions of unit norm. The solution obtained has norm bounded by a constant independent of the points chosen. As a direct application, we obtain a characterization of the order-boundedness of a sum of products of weighted composition and differentiation operators acting between weighted Bergman spaces. We also characterize the compactness of such operators that map a wei…
We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results …
The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K. Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames …
We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of…
Multiresolution in the Bergman space
本文在Bergman空间中构建多分辨率分析，通过一组极点生成多分辨率结构，并研究其密度、插值和采样性质，实现了Bergman空间信号的双曲小波表示。
Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
本文在多复变量加权Fock空间中，通过正则化曲率假设建立Bergman核估计，并构造\bar\partial解算子，进而用加权IDA空间刻画Hankel算子的有界性与紧性。
Convexity of the embedding parameter sets of some analytic function spaces
研究加权Bergman-Orlicz空间连续嵌入参数集的几何结构，证明在特定对数凸性和对数凹性条件下，可接受的权重指数集为凸集，并探讨其对偶问题。
It was shown by James Tung in 2005 that if a sequence $Z=\{z_n\}$ of points in the complex plane satisfies $$\inf_{n\not=m}|z_n-z_m|>2/\sqrtα,$$ then $Z$ is a sequence of interpolation for the Fock space $F^p_α$. Using results from circle packing, we show that the constant above can be improved to $$\sqrt{2π/(\sqrt3\,α)},$$ which is strictly smaller than $2/\sqrtα$. A similar result will also be obtained for sampling sequences.
Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert…
The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure o…
Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
本文为高斯混合范数Fock空间建立了采样与插值的尖锐判据，以经典密度α/π为临界值，并给出有界线性右逆，推广了经典Hilbert Fock空间理论。
Sampling constants in generalized Fock spaces
该研究证明了广义加倍Fock空间中支配集的Logvinenko-Sereda型定理，给出了采样常数对相对密度参数$γ$的精确多项式依赖，方法基于Remez型不等式和覆盖引理。
Beurling density theorems for sampling and interpolation on the flat cylinder
本文研究了平圆柱上的Fock空间，利用适应其几何结构的上下Beurling密度，完整刻画了采样集与插值集，并将结果等价转化为带时间周期化高斯窗的Gabor框架与Riesz基序列的完整刻画。
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