Derivative of functions in de branges and dirichlet spaces

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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.