巴拿赫空间的Feichtinger猜想、$R_\varepsilon$ -猜想和Weaver猜想
Feichtinger Conjectures, $R_\varepsilon$-Conjectures and Weaver's Conjectures for Banach spaces
摘要：
基于二十年前著名的框架的Feichtinger定理，希尔伯特空间的$R_\varepsilon$ -猜想和Weaver猜想（以及Marcus，Spielman和Srivastava的解答），我们给出了p-近似Schauder框架的Feichtinger定理，巴拿赫空间的$R_\varepsilon$ -猜想，Weaver猜想和Akemann-Weaver猜想.基于Casazza关于希尔伯特空间的框架的结果，我们也给出了p-近似Schauder框架的构造.基于希尔伯特空间框架的基本不等式和希尔伯特空间框架的标度问题，给出了p-近似Schauder框架的结构和问题.基于希尔伯特空间中Riesz基的Kothe-Lorch定理，给出了巴拿赫空间中p-逼近Riesz基的一个问题.我们构造了巴拿赫空间p-近似Schauder框架的动态采样问题。本文研究了巴拿赫空间p-近似Schauder框架的相位恢复问题和范数恢复问题。我们还制定了连续p-近似Schauder框架的离散化问题。
英文摘要：
Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality for frames for Hilbert spaces and scaling problem for Hilbert space frames. Based on Kothe-Lorch theorem for Riesz bases for Hilbert spaces, we formulate a problem for p-approximate Riesz bases for Banach spaces. We formulate dynamical sampling problem for p-approximate Schauder frames for Banach spaces. We ask phase retrieval problem and norm retrieval problem for p-approximate Schauder frames for Banach spaces. We also formulate discretization problem for continuous p-approximate Schauder frames.
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