Representation Of Holomorphic Functions Research Articles

Herglotz's representation and Carathéodory's approximation

AbstractHerglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

Solution of Problems of Elasticity Theory for Multiply Connected Half-Planes and Strips

A general solution of problems of elasticity theory for anisotropic half-planes and strips with arbitrary holes and cracks is presented, which uses the complex potentials of the plane problem of the theory of elasticity of an anisotropic body, conformal mappings, representations of holomorphic functions by Laurent series, and satisfaction of boundary conditions by the generalized least squares method. The problems are reduced to overdetermined systems of linear algebraic equations solved by the singular value decomposition. The results of numerical studies are described for a strip with a circular hole under its tension or under the action of a uniform pressure along a segment of a rectilinear boundary, as well as for the tension of a strip with a circular hole and a crack in the bridge, including those extending to the border of the strip or to the contour of the hole. An isotropic half-plane and a strip with holes and cracks are considered as special cases of the general problem. The influence of the geometric characteristics of holes and cracks, the physical and mechanical properties of the strip on the values and distribution of stresses material was studied.

Symmetric representations of holomorphic functions

Symmetric representations of holomorphic functions

Integral representations of holomorphic functions with a fixed point

Integral representations of holomorphic functions with a fixed point

On the representation of holomorphic functions on polyhedra

We generalize the Oka extension theorem, and obtain bounds on the norm of the extension, by using operator theory.

Sufficient sets in weighted Fréchet spaces of entire functions

Under study are sufficient sets in Frechet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for a weighted Frechet space. Applications are given to the problem of representation of holomorphic functions in a convex domain with certain growth near the boundary by exponential series.

Representation of holomorphic functions by schlömilch’s series

Abstract A necessary and sufficient condition is given for holomorphic functions to be represented by series of the kind $\sum\limits_{n = 0}^\infty {a_n J_0 (nz),z,a_n \in \mathbb{C},} $ where J 0(z) is the Bessel function of first kind with zero index. To derive the result, we use an Erdélyi-Kober operator of fractional order.

Integral representations of holomorphic functions

Integral representations of holomorphic functions

A topological version of the argument principle and Rouche’s theorem

In this paper, the topological nature of the generalization of the classical argument principle well-known in multidimensional complex analysis is discussed. The topological approach offered here ensures some topological results on the structure of pole and zero sets of holomorphic maps of bounded domains in complex manifolds. Some connections with integral representations of holomorphic functions are studied and a geometric interpretation of the Martinelli-Bochner complex-valued, differential-form realization is given.

Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

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 will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.

On the Rubel–Taylor Problem on a Representation of Holomorphic Functions

On the Rubel–Taylor Problem on a Representation of Holomorphic Functions

On the representation of holomorphic functions by integrals

Sufficient conditions are given for the holomorphy of functions defined by parametric integrals in Lebesgue integration spaces

Bochner-Martinelli formula with discrete holomorphic kernel

IT is well known that it is impossible to construct an integral representation of holomorphic functions with holomorphic kernel for the general bounded domain. In this note we establish a global integral formula of Bochner-Martinelli type with discrete holomorphic kernels; this formula may be applied to solving (?)-equation and singular integral equations....

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Frechet spaceE has the property (DN) if and only if every holomorphic function onE *, the strongly dual space ofE, with values in the strongly dual space of a Frechet spaceF having the property ( $$\bar \Omega$$ ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC ∞, the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE * does not have such a system whenE is a nuclear Frechet space that does not have the property (DN).

The property $(H_u)$ and $(\tilde\Omega)$ with the exponential representation of holomorphic functions

The main aim of this paper is to prove that a nuclear Frechet space $E$ has the property $(H_u)$ (resp. $(\tilde\Omega)$) if and only if every holomorphic function on $E$ (resp. on some dense subspace of $E$) can be written in the exponential form.

The property $(H_u)$ and $(\tilde\Omega)$ with the exponential representation of holomorphic functions

The property $(H_u)$ and $(\tilde\Omega)$ with the exponential representation of holomorphic functions

Representation of Holomorphic Functions by Boundary Integrals

Let $K$ be a compact locally connected set in the plane and let $f$ be a function holomorphic in the extended complement of $K$ with $f(\infty ) = 0$. We prove that there exists a sequence of measures $\{ {\mu _n}\}$ on $K$ satisfying ${\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0$ such that $f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }$. It follows from the proof that two topologies for the space of functions holomorphic on $K$ are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on $K$. A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when $K$ is a compact subset of the unit circle which is not locally connected.

Representation of holomorphic functions by boundary integrals

Let K K be a compact locally connected set in the plane and let f f be a function holomorphic in the extended complement of K K with f ( ∞ ) = 0 f(\infty ) = 0 . We prove that there exists a sequence of measures { μ n } \{ {\mu _n}\} on K K satisfying lim n → ∞ | | μ n | | 1 / n = 0 {\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0 such that f ( z ) = ∑ n = 0 ∞ ∫ K ( w − z ) − n − 1 d μ n ( w ) ( z ∈ K ) f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} } . It follows from the proof that two topologies for the space of functions holomorphic on K K are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on K K . A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when K K is a compact subset of the unit circle which is not locally connected.

Integral Representations of Holomorphic Functions

Integral Representations of Holomorphic Functions