Let D, be a homogeneous bounded domain in Cl’ containing the origin 0. Let G, be the group of holomorphic transformations of D, and let us consider a representation T, of G inside a Hilbert space H, of holomorphic functions on D, . As a simple example, suppose that D,-, = {D, n z, = 0} is a homogeneous bounded domain in P-l for a subgroup G,_, of G, . We consider the restriction of the represen...

Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in [12] like a higher dimensional analogue of finely holomorphic functions.

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.

Let G(xr − xe) be the causal Green function for the wave equation in four spacetime dimensions, representing the signal received at the spacetime point xr due to an impulse emitted at the spacetime point xe. Such emission and reception processes are highly idealized, since no signal can be emitted or received at a single (mathematical) point in space and time. We present a simple model for exte...

In this paper we obtain a noncommutative multivariable analogue of the classical NevanlinnaPick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function f : Λ → C, where Λ is an arbitrary subset of the open unit ball Bn := {z ∈ Cn : ‖z‖ < 1}, we find necessary and sufficient conditions for the existence of a free holomorphic function g with c...

We first extend a classical iteration formula of holomorphic functions to higher dimensional holomorphic mappings. Then as an application, we prove Poincare’s center theorem about the condition under which a singularity of a holomorphic vector field is an isochronous center. 1. An iteration formula Let f be a holomorphic function germ at the origin 0 in the complex plane with the Taylor expansion

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space X is the unit sphere SX if X is locally c-convex. In particular, it is shown that the unit sphere of the OrliczLorentz sequence space λφ,w i...

We construct the so-called theta vectors on noncommutative T , which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions, the theta vectors. The holomorphic structure in the noncommutative T 4 case is given by a 2×2 co...

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper we construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.