Derivative of functions in de branges and dirichlet spaces

复分析和谐波分析是古典分析中两个古老而密切相关的分支。此外，它们在应用科学和工程中有许多应用。例如，很难找到一个没有听说过傅里叶分析的工程师。Hardy空间H2的模型子空间是我之前NSERC研究项目的主题。毫无疑问，这门学科在纯科学和应用科学中有各种各样的应用，例如算子理论、控制工程和量子物理。V. Havin和我在《加拿大数学杂志》上发表了两篇关于这个主题的长篇论文，因此获得了加拿大数学学会的G. de . B. Robinson奖。德布朗日空间是模型子空间的推广。狄利克雷空间是解析函数的另一类空间，与哈代空间关系密切。研究这些空间是一个新的、高度活跃的研究领域，有许多有趣的开放性问题。特别是，我们想探索这些空间在控制理论中的应用，这是一个尚未触及的主题。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。本课题研究了这些空间中函数的导数、平移算子作用下的不变子空间及其零集。

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.