Mishko Mitkovski, Brett D. Wick
It was known to von Neumann in the 1950's that the integer lattice $\mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallstén that a sequence of points $Γ$ that is uniformly close to the integer lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.
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We characterize zero sets for which every subset remains a zero set too in the Fock space $\mathcal{F}^p$, $1\leq p<\infty$. We are also interested in the study of a stability problem for some examples of uniqueness set with zero excess in Fock spaces.
We study the uniqueness sets, the weak interpolation sets, and convergence of the Lagrange interpolation series in radial weighted Fock spaces
We study multiple sampling, interpolation and uniqueness for the classical Fock space in the case of unbounded mul-tiplicities.
Phase retrieval in Fock space and perturbation of Liouville sets
本文研究 Fock 空间中的相位恢复问题，通过构建 Liouville 集合的扰动作为不规则采样集，首次实现了具有有限密度的唯一性集合构造，并将结果推广至 Gabor 相位恢复问题。
Asymmetric uniqueness sets in $\ell^q$
本文研究了$\ell^q$中唯一集的非对称现象，揭示了单边与双边$\ell^q$唯一性问题之间的显著差异。
Uniqueness sets in weighted $\ell^2$-spaces via simultaneous approximation
本文从同时逼近视角研究加权 $\ell^2$ Fourier 空间中的唯一性集合，建立位势理论框架以统一经典结果，并揭示单侧与双侧唯一性集合的深刻差异。
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.
We prove that zero sets for distinct Fock spaces are not the same, this is an answer of a question asked by K. Zhu in \cite[Page. 209]{Zhu}.
We consider inductive limits of weighted spaces of holomorphic functions in the unit ball of $\mathbb C^n$. The relationship between sets of uniqueness, weakly sufficient sets and sampling sets in these spaces is studied. In particular, the equivalence of these sets, under general conditions of the weights, is obtained.
On the zero sets of the Bargmann Fock space
本文研究Bargmann–Fock空间在临界密度下精确零点集的刻画问题，证明经典密度理论失效，并提出基于重整化对数势和相对熵的变分判据，为临界构型提供尖锐条件。
Uniqueness of meromorphic functions that share Two Sets
本文引入新型有限值域集合对，研究亚纯函数由两个有限分担集合唯一确定的问题。
In this paper, we prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share three sets. The obtained results improve some recent existing results.
Given a nondecreasing sequence $Λ=\{λ_n>0\}$ such that $\displaystyle\lim_{n\to\infty} λ_n=\infty,$ we consider the sequence $\mathcal N_Λ:=\left\{λ_ne^{iθ_n},n\in\,\mathbb N\right\}$, where $θ_n$ are independent random variables uniformly distributed on $[0,2π].$ We discuss the conditions on the sequence $Λ$ under which $\mathcal N_Λ$ is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence $Λ$ with respect to the weight is found.
In this paper we have mainly focused on the uniqueness of two meromorphic functions under the realm of two shared sets problem with certain restrictions which in turn improve the results of { P. Sahoo and H. Karmakar, Uniqueness theorems related to weighted sharing of two sets, Acta Univ. Sapientiae, Mathematica, 10(2) (2018), 329-339} and {P. Sahoo and A. Sarkar, On the uniqueness of meromorphic functions sharing two sets, Bol. Soc. Mat. Mex., (2019), 1-7}. In particular, pointing out some gaps in the statements of the results in { P. Sahoo and H. Karmakar, Uniqueness theorems related to weig…
We give an extension of Poincaré's type capacitary inequality for Dirichlet spaces and provide an application to study the uniqueness sets on the unit circle for these spaces.
In this paper we prove a number of results concerning uniqueness of a meromorphic function as well as its derivative sharing one or two sets. In particular, we deal with the specific question raised in [18], [19], [20] and ultimately improve the result of Banerjee-Bhattacharjee [4].
Uniqueness sets with angular density for spaces of entire functions, II: $ρ=1$
本文研究指数型整函数空间中具有角密度的唯一性集合，在放弃Lindelöf条件限制的情况下展开分析，并保持内容自足可独立阅读。
In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate the uniqueness of zero order meromorphic function with its q-difference operator sharing two sets with finite weight. Some examples have been exhibited which are relevant to the content of the paper.
Uniqueness sets with angular density for spaces of entire functions, III: how to minimize the type
本文研究了整个函数空间中具有角密度的唯一性集，通过分析满足一定正则性条件的离散集Λ，证明存在一个类型最小化的测度Δ₀，其质量少于2ρ。对于ρ=2的情况，确定了临界唯一性类型的几何表达。
In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively. Moreover, we exhibit some examples to show the sharpness of some conditions used in our main result.
In this paper, we investigate the uniqueness property of meromorphic functions together with its linear difference polynomial sharing two sets. Using the polynomial introduced in [FILOMAT 33(18)(2019), 6055-6072], we have improved the result of Li-Chen [Abstract and Applied Analysis, 2014, Article ID 894968] in sense of reducing cardinalities of the main set S and the associated weights. Some examples have been exhibited to validate our certain claims in the main result.
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.
Let E be a compact set of C of positive logarithmic capacity. Let us suppose that for every polynomial $P\not=id$ we have $P^{-1}(E)\not=E$. Then for all no constant polynomials f and g such that $f^{-1}(E)=g^{-1}(E)$ we have f=g.
Uniqueness sets with angular density for spaces of entire functions, I. Basics
本文研究了具有角密度的整个函数空间的唯一性集，确定了临界零集类型并给出了其与临界唯一性集类型相吻合的条件，同时探讨了其在福克型空间中的应用。
In this article are improved and refined some of our results prior to our recent article from the journal "Russian Mathematics (Izvestiya VUZ. Matematika)" in 2015 at the expense of our latest results in 2016 on lower estimates of subharmonic functions by the logarithm module nonzero holomorphic function.
With the help of the notion of weighted sharing of sets, this paper dealt with the question posed by \emph{Yi} \cite{Yi-SC-1994} regarding the uniqueness of meromorphic functions concerning three set sharing. A result has been proved which significantly improved the recent results of {Banerjee - Ahamed} [Bull. Polish Acad. Sci., {62}(3)(2014), 243--256.], \{Banerjee - Mukherjee} [Hokkaido Math. J., {37}(2008), 507--530.] and {Banerjee - Majumder} [Analysis, {34}(2)(2014), 143--162.] by relaxing the nature of sharing. Several examples have been exhibited to show the sharpness of the cardinaliti…
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。
Uniqueness sets for functions of Dirichlet-type with restricted Taylor coefficients
研究 Dirichlet 型空间中由特定单项式张成的子空间的唯一性对，证明了若指数集包含足够长的等差序列则边界点快速趋近时无唯一性对，反之若指数集稀疏则可构造任意快速趋近边界的唯一性对。
Let $μ$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(μ)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. We give a sufficient condition on a Borel subset $E$ of the unit circle which ensures that $E$ is a uniqueness set for $\mathcal{D}(μ)$. {We also give somes examples of positive Borel measures $μ$ and uniqueness sets for $\mathcal{D}(μ)$.}
We show that under mild conditions, a Gaussian analytic function $\boldsymbol F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock space has the property that a.s. no non-zero function in that space vanishes where $\boldsymbol F$ does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann-Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro (1979) on Bergman spaces and allowing us to strengthen a result of Zh…
The purpose of the paper is to rectify a series of errors occurred in [2], [17], [20] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set [3] to settle a long time unsolved problem.
We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed unit disc in $\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\bar{D}^I)$ (respectively for $A(\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\partial \bar{B}_n$) where $T=\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. Finally,…
We introduce and study some new spaces of holomorphic functions on the right half-plane. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the M"untz--Sz'asz problem for the Bergman space, that is, the problem to characterize the sets of complex powers that form a complete set the unweighted Bergman space of a disc. In this paper, we construct a space of holomorphic functions on the right half-plane, whose sets of uniqueness correspond exactly to the sets of powers that are a complete set in Bergman space. We show that this space is a reproducing kernel Hilbert s…
Cartan uniqueness theorem on nonopen sets
本文研究非开集上的 Cartan 唯一性定理，通过解析圆盘包络条件建立全纯限制的唯一延拓，并推广至实解析子簇的无穷小 CR 自同构，证明圆形子簇的相关自同构必为线性。
Starting with a question of Yuan-Li-Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58(2)(2018), 249-262] we have studied the uniqueness of a meromorphic function f and an L-function L sharing two finite sets. At the time of execution of our work, we have pointed out a serious lacuna in the proof of a recent result of a of Sahoo-Halder [ Some results on L-functions related to sharing two finite sets, Comput. Methods Funct. Theo., 19(2019), 601-612] which makes most of the part of the Sahoo-Halder's paper under question. In context of our choice …
The purpose of the paper is to study the uniqueness problem of a $L$ function in the Selberg class sharing one or two sets with an arbitrary meromorphic function having finite poles. We manipulate the notion of weighted sharing of sets to improve one result of Yuan-Li-Yi [Value distribution of $L$-functions and uniqueness questions of F. Gross, Lithuanian Math. J., $\mathbf{58(2)}$(2018), 249-262]. More importantly, we have pointed out a number of gaps in all the results of Sahoo-Halder [Results on $L$ functions and certain uniqueness question of Gross, Lithuanian Math. J., $\mathbf{60(1)}$(20…
Identification of L-functions in the extended Selberg class by preimages of finite sets
本文改进了L函数在扩展Selberg类中的唯一性定理，去除了函数方程相同的要求，证明了多项式在某些条件下具有强唯一性。
This paper studies the uniqueness of two non-constant meromorphic functions when they share a finite set. Moreover, we will give the existence of unique range sets for meromorphic functions that are zero sets of polynomials that do not necessarily satisfy the Fujimoto's hypothesis.
Reverse Carleson inequalities for weighted Fock spaces with $A_{\infty}$-type weights
通过刻画Muckenhoupt $A_{\infty}$权诱导的加权Fock空间$F^p_{\alpha,w}$的控制集，给出了受限$A_{\infty}$权下$F^p_{\alpha,w}$的逆Carleson测度的充分条件，并应用于Toeplitz算子的可逆性、Volterra算子的闭值域性质及$F^p_{\alpha,w}$的原子分解。
Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations
本文探讨复值再现核希尔伯特空间中的代表定理，应用于福克空间、哈代空间及超振荡问题，揭示超振荡与机器学习核方法的联系。
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