Derivative of functions in de branges and dirichlet spaces

复变分析和调和分析是经典分析的两个古老而又密切相关的分支。此外，它们在应用科学和工程中也有许多应用。例如，很难找到一个没有听说过傅立叶分析的工程师。Hardy空间H2的模型子空间是我之前NSERC研究项目的主要主题。毫无疑问，这门学科在纯科学和应用科学中有着广泛的应用，例如算符理论、控制工程和量子物理。由于在《加拿大数学杂志》上发表了两篇长篇论文，V.Havin和我获得了加拿大数学学会的G.de B.Robinson奖。De Brange空间是模型子空间的推广。Dirichlet空间是与Hardy空间关系密切的另一族解析函数空间。研究这些空间是一个新的、非常活跃的研究领域，有许多有趣的开放问题。特别是，我们想要探索这些空间在控制理论中的应用，这是一个还没有被触及的主题。要做到这一点，我们必须丰富学科本身，并获得一些必要的工具。在这个项目中，我们研究了函数在这些空间中的导数，它们在移位算子下的不变子空间以及它们的零集。

Complex analysis and harmonic analysis are two old and closely related branches of classical analysis. Moreover, they have many applications in applied sciences and engineering. For example it is rather impossible to find an engineer who has not heard of Fourier analysis.Model subspaces of the Hardy space H2 was the main theme of my previous NSERC research project. No doubt this subject has a variety of applications in pure and applied sciences, e.g. operator theory, control engineering and quantum physics. V. Havin and I were awarded the G. de B. Robinson prize of the Canadian Mathematical Society for publishing two long papers in the Canadian Mathematical Journal on this subject.De Branges spaces are a generalization of model subspaces. Dirichlet spaces are another family of spaces of analytic functions with close relations to Hardy spaces. Studying these spaces is a new and highly active area of research with many interesting open questions. In particular, we would like to explore the application of these spaces in control theory, a subject that has not been touched yet. To do this we have to enrich the subject itself and obtain some necessary tools. In this project we study the derivative of functions in these spaces, their invariant subspaces under the shift operators and their zero sets.