The Inner Functions And Outer Functions Of The Dirichlet Spaces

This article introduces the concept of the inner functions and outerfunctions of the Dirichlet spaces (diferent from the Hardy spaces). Wefirst prove the inner functions and the extremal functions are equivalentin the Dirichlet spaces,and the outer functions are the functions of theintersection of the Hardy spaces and the Dirchlet spaces. Finally, wediscuss the factorization theory of the Dirichlet spaces, and we provedthat for any polynomial of the Dirichlet spaces can be factorized into theproduct of the inner function (extremal function) and cyclic vector.