Approximation Theory In Holomorphic Function Spaces

The main theme of the theory of function approximation is to approximation complicated functions with simple functions (such as algebraic polynomials, trigono-metric polynomials, and spline functions) and to study qualitative and quantitative properties of the approximation. There exist a lot on approximation theory for real variable functions as well as for one complex variable functions[76,77]. However, the theory of approximation for several complex variables has not yet been established.The topic of this dissertation is to study the theory of approximation for several complex variables. We introduce new type of holomorphic function spaces Qμas well as Aμassociated with measureμto unify of the existing many function spaces, such as BMOA, Bloch, Qp, Hardy, Bergman, Lipschitz, QK, F(p,q,s), ball-algebra, Bargmann spaces. We establish the Jackson theorem and the Bernstein theorem in Qμand Aμspaces, which yield the approximation results in many function spaces. We would like to point out that there is no result on the Bernstein theorem for several complex variables before our work. We also introduce K-functions in the study of the theory of approximation for several complex variables and with it as a tool, we establish in Qp spaces the strong inverse inequality and the weak equivalence theory between approximation by the Riesz operator and polynomial approximation. We generalize the Hardy-Littlewood theorem for Hardy spaces related to the smoothness of the boundary value functions to the case of Bergman spaces, which overcome the problem that in Bergman spaces can be no boundary values in difference to the case of Hardy spaces. This provides a new theory to extend the boundary value theory of Hardy spaces to Bergman spaces. We also extend recent result by Savchuk on Fejer approximation for Dirichlet functions from the unit disc to the unit ball.The concrete content of each chapter is as follows:The Jackson direct theorem is about how to control approximation of functions by the smoothness of functions, such as continuity, differentiability, and Lipschitz smoothness. We will establish the Jackson theorem in many spaces such as BMOA, Qp, Bloch, Besov, D-algebra. Moreover, we even establish the Jackson theorem for higher order derivatives in any starlike circular domain. With a unified approach, we establish the Jackson theorem for many holomorphic function spaces by introducing Qμand Aμspaces as a unified framework. (see Ch 2)The Bernstein theorem deals with the problem of how to control the smoothness of functions by the order of approximation. We establish the Bernstein inequality in many function spaces using a unified approach and, furthermore, we establish the Bernstein theorem. As an application of the Jackson theorem and the Bernstein theorem, we obtain an equivalence characterization of the Lipschitz subclass as well as the Zygmund subclass in the Qp spaces in terms of order of approximation by polynomials, (see Ch 3)In addition to the smoothness modulus, K-functional is another important tool in the theory of approximations. We introduce K-functionals in Qp spaces over the unit ball and obtain the strong inverse inequality, the weak equivalence relation between approximation by the Riesz operator and polynomial approximation, and the Marchaud inequality. (see Ch 4)The Hardy-Littlewood theorem for the Hardy spaces on the unit disc provides the close relation between the smoothness of the boundary values and the mean values of the derivatives. We shall extend this result to Bergman-type spaces on bounded symmetric domains. (see Ch 5)For the Dirichlet functions in the unit ball, we establish a precise estimate of approximation by Fejer operator with a Hardy space as an ambient space. (see Ch 6)Our theory on approximation in the case of several complex variables enriches and complements the existing function theory in Cn. Taking into account the impor-tance of approximation theory in real analysis, complex analysis, harmonic analysis, computational science, engineering, and information theory, we can expect wide applications of our theory in the future.