Interpolating and sampling sequences for entire functions

Pade interpolation by F-polynomials and transfinite diameter

We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly approximating $F$-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set $K\subset\mathbb C$ then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfi…

Sampling sets for Hardy spaces of the disk

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.

Interpolation and sampling in small Bergman spaces

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.

Fock空间中多项式的Marcinkiewicz-Zygmund不等式

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。

对小Fock空间中完整插值序列的统一方法

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。

Interpolation and Sampling in Analytic Tent Spaces

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.

Bandlimited Lipschitz functions

We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of the natural interpolating and sampling sequences for this space. We also find a description of the trace of such functions to sequences of critical density in terms of a cancellation condition.

Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

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.

Interpolation and Sampling Hypersurfaces for the Bargmann-Fock space in higher dimensions

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…

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

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…

Sampling and interpolation in de Branges spaces with doubling phase

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 in spaces of entire functions of exponential type in $\mathbb C^{n+1}$

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary $b\mathcal U$ of the Siegel upper half-space $\mathcal U$ and it is fundamental that $b\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. We consider entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathca…

双曲正割型函数的非规则采样

Irregular sampling for hyperbolic secant type functions

本文研究了双曲正割型窗函数下的Gabor框架，提出了半非规则采样的判据，并推广了已有结果，同时给出了由该函数生成的平移不变空间的完全插值序列的完整刻画。

关于方格上的离散全纯Paley-Wiener空间与采样

On discrete holomorphic Paley-Wiener spaces and sampling on the square lattice

本文研究了方格上离散整函数的再生核Hilbert空间，受经典Paley-Wiener空间启发，给出了该空间的Paley-Wiener型刻画与采样结果。

不规则采样下的解析性与超移

Analyticity and supershift with irregular sampling

本文研究不规则采样下超移与全实轴实解析性的关系，证明整函数在实数轴上的限制必有超移，并给出周期函数情形下规则采样逆命题的证明。

拟平移不变空间中的采样与由指数多项式比值生成的Gabor框架

Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

本文引入两类具有周期性的整函数与亚纯函数生成元族，涵盖高斯与双曲正割情形，证明了分离集密度的尖锐采样定理，并应用于半正则格点的Gabor框架密度新结果。