Problems in Function Theory and Operator Theory

翻译后摘要：申请人提出的几个相关问题，在functiontheoretic算子理论和欧几里德调和分析的工作。他计划利用他最近的工作与Arcozzi和索耶作为基础的研究插值序列在Besov空间。他建议使用相空间方法来研究特征向量的pseudodifferential运营商以及解决其他极值问题。他还计划使用他最近的工作与荷兰的渐近行为的伯格曼核函数作为一个出发点，以调查是否广义Segal-Bargmannspaces可以用来给量子变形的飞机。最后，他计划调查是否某些技术，出现在最近的工作高阶汉克尔算子有更广泛的应用在operatortheory。所提出的工作将促进对全纯函数和欧几里德调和分析中产生的算子空间的泛函理论算子理论的理解。它还将发展新的观点和技术，将在thoseareas具有普遍适用性。更一般地说，拟议的工作是在接口的运营商理论，函数理论和谐波分析。 工作的统一主题是相空间方法的系统使用，既作为一种技术工具，更重要的是，作为一个统一的概念结构。 这些工具和观点已经非常富有成效，无论是历史上还是最近，在理论数学中的重要新技术和数学在科学和工程中的主要新应用。 Lacey和Thiele最近关于双线性算子的工作是第一个例子，小波和相关方法对信号处理的影响是第二个例子。 人们当然期望在这些领域的进一步研究将继续在数学内外产生重要的额外贡献。

Abstract:The applicant proposes to work on several related problems in functiontheoretic operator theory and in Euclidean harmonic analysis. He plans touse his recent work with Arcozzi and Sawyer as a basis for the study ofinterpolating sequences in Besov spaces. He proposes to use phase spacemethods to study eigenvectors of pseudodifferential operators and as wellas solutions to other extremal problems. He also plans to use his recentwork with Holland on the asymptotic behavior of Bergman kernel functionsas a starting point to investigate whether generalized Segal-Bargmannspaces can be used to give quantum deformations of the plane. Finally, heplans to investigate whether certain techniques that arose in recent workon higher order Hankel operators have broader application in operatortheory. The work proposed will advance the understanding of the functiontheoretic operator theory on spaces of holomorphic functions and ofoperators arising in Euclidean harmonic analysis. It will also develop newviewpoints and techniques that will have general applicability in thoseareas.More generally, the proposed work is at the interface of operator theory,function theory, and harmonic analysis. The unifying theme of the work isthe systematic use of phase space methods, both as a technical tool and,more importantly, as a unifying conceptual structure. Those tools andthat viewpoint have been very productive, both historically and recently,of important new techniques in theoretical mathematics and of major newapplications of mathematics to science and engineering. The recent workLacey and Thiele on bilinear operators is an example of the first and theimpact of wavelet and related methods on signal processing is aspectacular example of the second. One certainly expects that furtherresearch in these areas will continue to produce important additionalcontributions inside and outside of mathematics.