This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

A new method for construction of bases for general distribution spaces is developed. This method allows the freedom to prescribe the nature and properties of the basis elements. The method is deployed to the construction of bases consisting of rational functions of uniformly bounded degrees for Besov and Triebel–Lizorkin spaces of holomorphic functions in the unit disc. In turn, this is utilize...

1. Holomorphic functions in one and several variables. Definition of a complex manifold. Canonical splitting of the complexified tangent and cotangent space. T R ⊂ T ⊗ C = T ′ ⊕ T ′′ , where T R is the real tangent space at a point, T ′′ = ∂/∂z i annihilates holomorphic functions and T ′ = ∂/∂z i annihilates antiholomorphic functions. On C, Df (w) = ∂f ∂z w + ∂f ∂z w. Quasiconformal maps. Stone...

I . Introduction There has been a large amount of study devoted to the problem of analytic continuation. If K is a subset of C" , 11 > l , then one wishes to know if there is a larger set K', containing K, such that all holomorphic functions on K can be extended to holomorphic functions on K'. It has been known for some time that the envelope of holomorphy of a domain in C" is a Stein manifold ...

We give characterizations of (quasi-)plurisubharmonic functions in terms Lp-estimates $$\overline{\partial}$$ and Lp-extensions holomorphic functions.

Let $Omega_X$ be a bounded, circular and strictly convex domain of a Banach space $X$ and $mathcal{H}(Omega_X)$ denote the space of all holomorphic functions defined on $Omega_X$. The growth space $mathcal{A}^omega(Omega_X)$ is the space of all $finmathcal{H}(Omega_X)$ for which $$|f(x)|leqslant C omega(r_{Omega_X}(x)),quad xin Omega_X,$$ for some constant $C>0$, whenever $r_{Omega_X}$ is the M...

Over the years the author has been interested in rigidity problems on bounded symmetric domains of rank ≥ 2. In this article we give an overview on rigidity problems arising from holomorphic mappings either on bounded symmetric domains of rank ≥ 2 or on their finite-volume quotient manifolds into complex manifolds, placing the focus on recent developments. The article highlights the use of some...

We define the differential operator ∂ ∂z on infinitely differentiable functions (also called smooth or C∞ functions) on some open set in C by ∂ ∂z = 1 2 ( ∂ ∂x + i ∂ ∂y ). A quick calculation shows that ∂ ∂z obeys the product rule. Recall that a function f is holomorphic if and only if ∂ ∂z (f) = 0. A function is biholomorphic, or an analytic isomorphism, if it is holomorphic and has holomorphi...

Drasin’s theorem describing meromorphic functions of finite order with maximal sum of deficiencies is extended to holomorphic curves in projective space. A conjecture about holomorphic curves extremal for Cartan’s defect relation is discussed.