Functions Of Many Complex Variables Research Articles

Approximate methods for solving amplitude-phase problem for discrete signals

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.

Solvability of a nonlocal boundary-value problem for the operator-differential equation with weak nonlinearity in a refined scale of Sobolev spaces

A nonlocal boundary-value problem for the differential equation with weak nonlinearity and with differential operator B = (B 1, …, B p ), where \( {B}_j\equiv {z}_j\frac{\partial }{\partial {z}_j}\;\mathrm{and}\;j=1,\dots, p \) is considered. By using the Nash–Moser iterative scheme, the solvability conditions for the present problem in the Hilbert H¨ormander spaces of functions of many complex variables forming a refined Sobolev scale of spaces is established.

Solvability of the Nonlocal Boundary-Value Problem for a System of Differential-Operator Equations in the Sobolev Scale of Spaces and in a Refined Scale

We study the solvability of the nonlocal boundary-value problem with one parameter for a system of differential-operator equations in the Sobolev scale of spaces of functions of many complex variables and in the scale of Hormander spaces which form a refined Sobolev scale. By using the metric approach, we prove the theorems on lower estimates of small denominators appearing in the construction of solutions of the analyzed problem. They imply the unique solvability of the problem for almost all vectors formed by the coefficients of the equation and the parameter of nonlocal conditions.

On the Best Polynomial Approximations of Entire Transcendental Functions of Many Complex Variables in Some Banach Spaces

For the entire transcendental functions f of many complex variables m (m > 2) of finite generalized order of growth ⇢m(f; ↵, β), we obtain the limiting relations between the indicated characteristic of growth and the sequences of best polynomial approximations of f in the Hardy Banach spaces Hq(U m ) and in the Banach spaces Bm(p,q, λ) studied by Gvaradze. The presented results are extensions of the corresponding assertions made by Varga, Batyrev, Shah, Reddy, Ibragimov, and Shikhaliev to the multidimensional case. 1. For functions of one complex variable, the history of the problem of relationship between the rate of convergence of the sequences of best polynomial approximations of the entire transcendental functions to zero and some generalized characteristics of their growth is fairly comprehensively presented in [1]. The number of available results for the entire transcendental functions of many complex variables is much smaller (see, e.g., [2–4]). In the present paper, we continue our investigation of this problem in the multidimensional case. To formulate the required notions and definitions, we introduce the following notation: C is a space of complex

The study of a generalized integral of the Temlyakov type in the unit bicircle

The theory of integral representations of analytic functions of many complex variables is an important branch of the multidimensional complex analysis. An essential contribution to this theory was done by A. A. Temlyakov, who obtained integral representations for functions of two complex variables [1, 2]. The functions were assumed to be analytic in the class of parametrically defined bounded convex complete bicircular domains. L. A. Aizenberg [3] considered an arbitrary function summable in the sense of Lebesgue at the boundary of the defining domain as the density in the Temlyakov integrals. On the base of Temlyakov integral representations he introduced the notion of Temlyakov type integrals. This paper is dedicated to the study of properties of a generalized integral of the Temlyakov type

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.

Error-bounds for balanced model-reduction of linear time-varying systems

Error-bounds are developed for balanced truncation of linear time-varying systems, leading to an extension of the twice the sum of the tail formulas, well known in the time-invariant case. The approach relies on an operator-theoretic framework for analysis of linear time-varying systems. This provides a multivariable notion of frequency for such systems, which are thus characterized by rational functions of many complex variables, allowing the problem to be formulated in the linear-fractional framework. Using a time-varying version of standard necessary conditions for reduced-order modeling, based on convex operator inequalities, we show that these error-bounds for balanced truncation are related to the closest possible reduced-order modeling error in a sense which parallels the time-invariant case.

On integrable systems and semiclassical solutions of the stationary Schrodinger equations

Stationary Schrodinger equations corresponding to some well known integrable problems are defined. Then semiclassical modes for these equations are constructed as functions of many complex variables on the Jacobi varieties of compact multisheeted Riemannian surfaces. Quantum conditions are defined as conditions of finiteness on the number of sheets in the Riemannian surface.

A unification of boson expansion theories: (I). Functional representations of fermion states

The method for obtaining boson expansion by representing the fermion states as holomorphic functions of many complex variables is presented. Such functional representation is explicitly constructed for each space which is the carrier space of an irreducible representation of a semisimple compact Lie group. This is achieved by proving the unity resolution in terms of holomorphically parametrized Perelomov's generalized coherent states. The functional images of fermion states are polynomials of complex variables, while those of fermion operators are differential operators of finite order with polynomial coefficients.

Holonomic functions in a polycircle with nonnegative imaginary part

A well-known theorem of Nevanlinna on the representation of nonnegative measure of a function holomorphic in a circle and having nonnegative imaginary part is extended to functions of many complex variables, holomorphic in a polycircle and having there a nonnegative imaginary part.

Majorant domains for functionals on families of holomorphic functions of many complex variables

Majorant domains for functionals on families of holomorphic functions of many complex variables

Special methods of analytic completion in field theory

By combining known theorems in the theory of functions of many complex variables and distribution theory a technique of analytic completion is developed; this provides a simple proof of the ‘edge of the wedge* theorem. An integral representation for the double commutator is derived, and it is used to simplify some of the work involved in computing the envelope of holomorphy for the vertex function. The Jacobi identities have not been incorporated, with the consequence that the threefold case cannot be completely solved by this method. The technique is applied to the fourfold (scattering) function, again without the Jacobi identities being included. By this technique analytic completion can be performed for some but not all the domains encountered in the fourfold problem.