Luis Alberto Escudero, Antti Haimi, José Luis Romero
We characterize sampling and interpolating sets with derivatives in weighted Fock spaces on the complex plane in terms of their weighted Beurling densities.
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Generalized Fock space and fractional derivatives with Applications to Uniqueness of Sampling and Interpolation Sets
本文提出了一种广义福克空间，并利用修改后的Bargmann变换研究了采样和插值集的唯一性问题。
We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice.
We show that de Branges-Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy-Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
In this paper, we define the quaternionic Fock spaces $\mathfrak{F}_α^p$ of entire slice hyperholomorphic functions in a quaternionic unit ball $\mathbb{B}$ in $\mathbb{H}.$ We also study growth estimate and various results of entire slice regular functions in these spaces. The work of this paper is motivated by the recent work of \cite{alpa12} and \cite{marco}.
Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a "Nyquist rate" which increases with $n$, the degree of polyanaliticity of the space. Such conditions are equivalent to sharp lattice density conditions for certain vector-valued Gabor systems, namely superframes and Gabor super-Riesz sequences with Hermite windows, and in the case of superframes they were studied recently by Gröchenig and Lyubarskii. The proofs of our main results use variations of the Janssen-Ron-Shen duality p…
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。
Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS
本文研究了向量值RKHS中的抽象Kramer采样定理，推广了拟Lagrange型插值概念，并探讨了其与广义后向平移算子不变条件及de Branges空间的联系。
Weighted superposition operators on Fock spaces
本文刻画了Fock空间间加权叠加算子存在的整函数对条件，并证明其无紧算子，主要贡献在于完整描述该算子的有界性与连续性。
In this paper, we study the quaternionic counterpart of complex Fock spaces $\mathfrak{F}_α^p ( 0<p<\infty$ and for some parameter $α$) of entire slice hyperholomorphic functions in an Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{H}^n.$ We also study some basic properties of these spaces.
We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.
We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density $D$ with the following property: a set of sampling has density $\geq D$, whereas a set of interpolation has density $\leq D$. The main theorem unifies many known d…
We consider the Fock-Sobolev space $F^{p,m}$ consisting of entire functions $f$ such that $f^{(m)}$, the $m$-th order derivative of $f$, is in the Fock space $F^p$. We show that an entire function $f$ is in $F^{p,m}$ if and only if the function $z^mf(z)$ is in $F^p$. We also characterize the Carleson measures for the spaces $F^{p,m}$, establish the boundedness of the weighted Fock projection on appropriate $L^p$ spaces, identify the Banach dual of $F^{p,m}$, and compute the complex interpolation space between two $F^{p,m}$ spaces.
We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.
The Fock space consists of all entire functions which are square integrable with respect to Gauss measure. The Toeplitz algebra is the C*-algebra generated by the Toeplitz operator with bounded symbol on the Fock space. In this paper, we study the Toeplitz algebra on the Fock space.
Boundedness and compactness of Hausdorff operators on Fock spaces
该研究刻画了不同指标Fock空间之间Hausdorff算子的有界性与紧性，推广了$F^\infty_α$上的紧性结果到更一般整函数Banach空间，并给出$p$-求和条件。
Sampling properties of the zeroes of the Gaussian entire function
本文研究Fock空间上高斯整函数零点的采样性质，通过放宽密度与分离条件并估计邻近零点数量，证明了使用d+o(d)个点的随机采样不等式及采样常数的次多项式增长，并恢复了零点几乎必然为唯一性集的结果。
Relations between two classes of Hilbert spaces of entire functions, de Branges spaces and Fock-type spaces with non-radial weights, are studied. It is shown that any de Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g., weights depending on the imaginary part only) the corresponding de Branges spaces are explicitly described.
Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space
本研究探究加权L²空间中多项式的Marcinkiewicz-Zygmund族与Fock空间整函数采样定理的关系及相关对偶关系，最终得到借助Gabor框架描述Hermite函数张成的信号子空间的方法。
We investigate necessary and sufficient conditions under which entire functions in de Branges spaces can be recovered from function values and values of derivatives. Our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half-plane.
We give a negative answer to the Newman--Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question.
We characterize the Schatten class membership of the canonical solution operator to $\bar\partial$ acting on $L^2(e^{-2ϕ})$, where $ϕ$ is a subharmonic function with $Δϕ$ a doubling measure. The obtained characterization is in terms of $Δϕ$. As part of our approach, we study Hankel operators with anti-analytic symbols acting on the corresponding Fock space of entire functions in $L^2(e^{-2ϕ})$.
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…
Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Car…
IDA symbols and Schatten--Lorentz theory for Toeplitz operators on weighted Fock spaces
本文研究加权Fock空间上Toeplitz算子的Schatten--Lorentz类成员资格，通过局部解析距离与Lorentz条件给出完整刻画，并推导出二次权情形的显式Weyl常数。
We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.
We obtain a complete characterization of the entire functions $g$ such that the integral operator $(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$ is bounded or compact, on a large class of Fock spaces $\mathcal{F}^ϕ_p$, induced by smooth radial weights that decay faster than the classical Gaussian one. In some respects, these spaces turn out to be significantly different than the classical Fock spaces. Descriptions of Schatten class integral operators are also provided. En route, we prove a Littlewood-Paley formula for $||\cdot||_{\mathcal{F}^ϕ_p}$ and we characterize the positive Borel measures for…
Absolutely summing Toeplitz operators on weighted Fock spaces
本文研究加权Fock空间上复符号Toeplitz算子的绝对r-求和性，通过IDA分解和Rademacher论证等工具，给出了基于局部均值、复球平均和Berezin变换的等价刻画，并证明了准则的最优性。
We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.
A new characterization of the Hardy space and of other spaces of analytic functions
本文通过导数算子的伴随算子特征化Hardy空间及其他相关的再生核Hilbert空间，并提出新的Fock空间特征化方法。
Algebras of entire functions and representations of the twisted Heisenberg group
本文在扭曲Fock空间上构造酉算子族，通过复合公式引出包含两个海森堡群副本的群，并研究其不可约表示及对实子群的限制性质。
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。
A uniform approach to complete interpolating sequences for small Fock spaces with $p > 0$
本文提出了一种统一方法，研究小Fock空间中完整插值序列的特性，揭示了双侧空间在不同$p$值下的周期性现象。
Holomorphic Interpolation of Multivariate Completely Monotone Functions
通过将完全单调函数表示为正测度的Laplace或Stieltjes-Fantappiè变换，利用非交换Radon变换框架结合矩阵束实现与Weyl运算微积，实现有限点插值，得到方向完全单调的整函数或有理函数。
New characterizations for Fock spaces
本文证明了极大Fock空间$F^\infty_α$是Lipschitz空间，并提出了$F^p_α$的Hardy-Littlewood类型刻画的新方法。
We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left…
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。
Positive Toeplitz operators on pluriharmonic Fock spaces: Schatten class criteria, sharp norm comparisons, and generalized weights
该研究针对多重调和Fock空间上的正Toeplitz算子，建立了其Schatten类准则，解决了相关猜想，得到了与对应全纯Toeplitz算子的精确范数比较，还获得了对称范数理想的精确比较及迹恒等式。
We confirm Flandrin's prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, …
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