Spaces of analytic functions in a region with an angle

In this paper we consider the space A_p of analytic functions which are p-power integrable in a region with an angle. We find a set of numbers p and q (1/p+1/q=1) (which depend on the magnitude of the angle) for which the spaces A_p and A_q are mutually conjugate. In each of these spaces we introduce the orthonormal system $$e_n = \sqrt {{{\left( {n + 1} \right)} \mathord{\left/ {\vphantom {{\left( {n + 1} \right)} \pi }} \right. \kern-\nulldelimiterspace} \pi }} \varphi \prime \varphi ^n ,n = 0,1, \ldots ,$$ where? is the conformal mapping of the region onto the unit disc. We prove it is dense and determine when it will be a basis.     

A Pochhammer type sampling reconstruction of analytic functions

We give a characterization of the functions f analytic on the half-plane Re(z)>σ that can be expressed in the form for some βn∈C. It will follow that the βn depend on the values of f on the integers so this gives a sampling reconstruction of these functions. Several examples of this representation are given.     

SPACES OF ANALYTIC FUNCTIONS REPRESENTED BY DIRICHLET SERIES OF TWO COMPLEX VARIABLES

We consider the space X of all analytic functions f(s1,s2)=∑∞m，n=1 amnexp(s1λm s2μn) of two complex variables s1 and s2,equipping it with the natural locally convex topology and using the growth parameter,the order of f as defined recently by the authors.Under this topology X becomes a Frechet space.Apart from finding the characterization of continuous linear functionals,linear transformation on X,we have obtained the necessary and sufficient conditions for a double sequence in X to be a proper bases.     

Spaces of analytic functions represented by Dirichlet series of two complex variables

We consider the space X of all analytic functionsf(s1 ,s2) = ∞∑aminexp(s1λm+s2μtn)of two complex variables s1 and s2, equipping it with the natural locally convex topology and using thegrowth parmeter, the order of f as defined recently by the authors. Under this topology X becomes aFrechet space. Apart from finding the characterization of continuous linear functiors, linear transforma-tion on X, we have obtained the necesary and sufficient conditions for a double sequence in X to be a properbases.     

Parabolic analytic functions

Let bev=x+αy where α#0, α² = 0,x andy real are elements of a commutative ring. So inR(1, 3) ife ₀,e ₁,e ₂,e ₃ form a canonical framee ₀²=1,e ₁²=e ₂²=e ₃²=−1 the vector α=e ₀+e ₁ is different from zero while

Summation of analytic functions

For analytic functions of a certain class we establish the existence of a regular matrix summing to f(z) in the whole domain of analyticity off(z). In particular, the class under consideration contains the function (1 – z)^–1.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 3–10, January, 1976.     

Generalization of analytic functions

The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy-Riemann equations for delta case.In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy-Riemann equations for nabla case.     

Growth of analytic functions

For a function f analytic in a disk {z ¦ z ¦ <R }, 0 <R + , a criterion is obtained, in terms of the Taylor coefficients of the function, for the validity of the relation (r) ln⁺ M (r)dr < + , whereM (r)= max {¦ (f (z) ¦ ¦z ¦=r < R}, and is a function positive on [0, R) and satisfying certain conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 668–673, May, 1990.     

On approximation and generalized type of analytic functions of several complex variables

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.