稀疏序列上的离散希尔伯特变换
Discrete Hilbert transforms on sparse sequences
摘要：
研究了从$\ell^2_v$到加权$L^2$空间的加权离散希尔伯特变换$(a_n)_n \mapsto \sum_n a_n v_n/(z-γ_n)$，其中$Γ=(γ_n)$是复平面上不同点的序列，$v=(v_n)$是相应的正数序列.在特殊情况下，当$|γ_n|$增长至少呈指数，有界变换的这种描述在一个简单的相对于Muckenhoupt $(A_2)$条件。当$z$被限制到另一个序列$Λ$的特殊情况下进行了详细的研究，它表明，满足一定的容许条件的有界变换可以被分裂成许多满射变换，并找到这样的两个重量变换可逆的精确几何条件。这些结果可以解释为某些Hilbert空间中再生核系统的陈述，其中de Branges空间和$H^2$的模型子空间是最好的例子。特别是，连接到Feichtinger猜想指出。描述Carleson措施和Riesz基地的正规化再生核的某些“小”de Branges空间遵循本文的结果。
英文摘要：
Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-γ_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $Γ=(γ_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|γ_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $Λ$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of $H^2$ are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
