Однородные уравнения типа π-свертки

This paper is devoted to spectral synthesis of the operator adjoint to multiplication by a power of the independent variable in weighted spaces of entire functions of a single complex variable, and is closely connected with equations of convolution type, and with the general theory of subspaces invariant under a multiple differentiation operator. The problem of approximating solutions of a homogeneous equation of -sided convolution type by elementary solutions is solved. Certain systems of such equations are investigated.

Let be a convex domain in the complex plane and the space of holomorphic functions in with the topology of uniform convergence on compact subsets of . Let and be a pair of (differentiation) invariant subspaces of admitting spectral synthesis. Conditions ensuring that the intersection also admits spectral synthesis are described. One consequence of these conditions is a recent result of Abuzyarova (in a new, constructive and quantitative setting) on the representation of an invariant subspace admitting spectral synthesis as the solution space of a system of two homogeneous convolution equations. New approximation results for entire functions of exponential type are used.

Let I be a closed submodule over a polynomial ring in a space of holomorphic functions on a domain in the complex plane. We establish sufficient conditions under which I is generated by two functions or two special submodules. As a corollary, it follows from these results that if an invariant subspace W ⊂ C∞(a,b) (with respect to the differentiation operator) admits spectral synthesis, then it is the solution space of a system of two homogeneous convolution equations.

The behavior of nature is usually modelled with Differential Equations in various forms. Depending on the constrains and the accuracy of a model, the connected equations may be more or less complicated. For simple models we may use Non Homogeneous Equations but in general, we have to deal with Homogeneous ones since from a physicists point of view nature seems to be Homogeneous. In many applications of sciences, for solving many of them, often appear equations of type nth order Linear differential equations, where the number of them is Euler-Cauchy differential equations. i.e. Euler-Cauchy differential equations often appear in analysis of computer algorithms, notably in analysis of quick sort and search trees; a number of physics and engineering applications. In this paper, the researcher aims to present the solutions of a homogeneous Euler-Cauchy differential equation from the roots of the characteristics equation related with this differential equation using MATLAB. It is hoped that this work can contribute to minimize the lag in teaching and learning of this important Ordinary Differential Equation.

A discrete version of complex Vekua type equations is considered. We construct a repre\-sentation formula for the solution of the homogeneous complex equation and investigate the inhomogeneous equation. Similar to the continuous case this representation formula is a product of two functions. We also factorize the solution of a homogeneous Vekua type equation in the quaternionic case. In the complex plane we analyse both factors in detail and study the relation between the non-holomorphic factors in the discrete and continuous case.

SECTION 6 - Solutions of Linear Equations

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

In this paper we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on a bounded domain of the complex plane.

Hadamard type operators on spaces of real analytic functions in several variables

The “solutions” of boundary-value problems presented in Section 2.8 are merely formal in that the appropriate Green functions must be found, and finding these Green functions is extremely difficult except in certain special cases such as the Rayleigh-Sommerfeld problem. Moreover, even when the appropriate Green functions can be found they are often expressed in the form of superpositions of elementary solutions of the homogeneous Helmholtz equation that are especially suited for dealing with boundaries of a specific shape. One example of a set of such elementary solutions is the plane waves which arise from applying the method of separation of variables to the homogeneous Helmholtz equation using a Cartesian coordinate system and, as we will see below, form a complete set of basis functions for fitting boundary-value data specified on plane surfaces; e.g., for the RS problems. However, the plane waves have limited utility in solving boundary-value problems involving non-planar boundaries such as spherical boundaries. In that case the method of separation of variables is applied to the Helmholtz equation using a spherical polar coordinate system and the so-called multipole fields arise as a set of elementary solutions that form a basis for fitting boundary-value data specified on spherical boundaries. In this chapter we will briefly review the method of separation of variables for the Helmholtz equation and obtain the resulting eigenfunctions for the important cases of Cartesian, spherical polar and cylindrical coordinate systems.

An effective method for solving boundary value problems for the Laplace and Helmoltz equations (in domains possessing a definite symmetry) is the method of separation of variables. The general idea of this method is to find a set of solutions of the homogeneous partial differential equation in question that satisfy certain boundary conditions. These solutions then serve as “atoms”, from which, based on the linear superposition principle, one constructs the “general” solution. Since each of these “atoms” is a solution of the corresponding homogeneous equation, their linear combination is also a solution of the same equation.KeywordsDirichlet ProblemPoisson EquationArbitrary ConstantElliptic ProblemLaplace EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dirac operators, Schrödinger type operators in [formula omitted]n, and Huygens principle

This article investigates residues associated with holomorphic mappings defined on a complex space . By means of a new definition of principal value of a residue, it sharpens results of Coleff, Herrera, and Dolbeault concerning the structure of residues. It establishes a connection between residues and functionals in orthogonal to the ideal . Using these results on residues and functionals, a formula is derived for the exponential representation for elements of invariant subspaces and for the solution of homogeneous convolution equations.

Approximation of solutions of a homogeneous convolution equation whose characteristic function satisfies lower estimates

Exponential polynomials, i.e. solutions of linear homogeneous differential equations with constant coefficients, are characterized as continuous solutions of linear homogeneous difference equations with constant coefficients. Namely, denote by C the \({\Bbb C}\)-linear space of all continuous functions on \({\Bbb R}\), and define for \( \omega \in {\Bbb R}_+\) the shift operator \(z_{\omega }:C\to C\) by \(\eta \mapsto z_{\omega }\eta \) such that \(z_{\omega }\eta (t)=\eta (t+{\omega })\) for all \(t\in {\Bbb R}\). Every ideal \({\mathfrak a}\) of \(R:={\Bbb C}[x,y]\) determines a subspace¶¶\( C_{\frak a}=\{\eta \in C:f(z_{{\omega }_1},z_{{\omega }_2})\eta =0\) for all \(f(x,y)\in {\frak a}\}\,.\)¶¶For incommensurable \({\omega }_1,{\omega }_2\in {\Bbb R}_+\) and ideals \({\frak a}\) in R with a factor ring \(R/{\frak a}\) , which is finite dimensional as a \({\Bbb C}\)-linear space, the set \(C_{\mathfrak a}\) is characterized. The main idea consists in applying the Kronecker approximation theorem to the analytic transition from the explicit representation of the solutions of recurrence sequences to that of the continuous solutions of difference equations. For an alternative proof, the Kronecker approximation theorem is combined with the Jordan decomposition theorem.