# Volume 3 Issue #1 (March 2011)

• Isaev K. P., Yulmukhametov R. S. Unconditional exponential bases in Hilbert spaces. Pp. 3 - 15
Abstract: In the present paper, we consider the existence of unconditional exponential bases in general Hilbert spaces $H=H(E)$ consisting of functions defined on some set $E\subset\mathbb C$ and satisfying the following conditions. 1. The norm in the space $H$ is weaker than the uniform norm on $E$, i.e. the following estimate holds for some constant $A$ and for any function $f$ from $H$: $\|f\|_H\le A\sup_{z\in E}|f(z)|.$ 2. The system of exponential functions $\{\exp(\lambda z),\lambda\in\mathbb C\}$ belongs to the subset $H$ and it is complete in $H$. It is proved that unconditional exponential bases cannot be constructed in $H$ unless a certain condition is carried out. Sufficiency of the weakened condition is proved for spaces defined more particularly.
• Napalkov V. V. On orthosimilar systems in a space of analytical functions and the problem of describing the dual space. Pp. 31 - 42
Abstract: We consider an orthosimilar system with the measure $\mu$ in the space of analytical functions $H$ on the domain $G\subset\mathbb C$. Let $K_H(\xi,t)$, $\xi,t\in G$, be a reproduction kernel in the space $H$. We claim that a system $\{K_H(\xi,t)\}_{t\in G}$ is the orthosimilar system with the measure $\mu$ in the space $H$ if and only if the space $H$ coincides with the space $B_2(G,\mu)$. A problem of describing the dual space in terms of the Hilbert transform is considered. This problem is reduced to the problem of existence of a special orthosimilar system in $B_2(G,\mu)$. We prove that the space $\widetilde B_2(G,\mu)$ is the only space with a reproduction kernel and it consists of functions given on the domain $\mathbb C\setminus\overline G$ with an orthosimilar system $\{\frac1{(z-\xi)^2}\}_{\xi\in G}$ with the measure $\mu$.
• Putintseva A. A. Riesz bases in weighted spaces. Pp. 47 - 52
Abstract: The article deals with weighted Hilbert spaces with convex weights. Let $h$ be a convex function on a bounded interval $I$ of the real axis. We denote a space of locally integrable functions on $I$, such that $\|f\|:=\sqrt{\int _I|f(t)|^2e^{-2h(t)}\,dt}<\infty$ by $L_2(I,h)$. If $I=(-\pi;\pi)$, $h(t)\equiv1$, the space $L_2(I,h)$ coincides with the classical space $L_2(-\pi;\pi)$ and the Fourier trigonometric system is a Riesz basis in this space. As it has been shown by B. J. Levin, nonharmonic Riesz bases in $L_2(-\pi;\pi)$ can be constructed using a system of zeros of entire functions of sine type. In this paper we prove that if a Riesz basis of exponentials exists in the space $L_2(I,h)$, this space is isomorphic (as a normed space) to the classical space $L_2(I)$. Thus, the existence of Riesz bases of exponentials is the exclusive property of the classical space $L_2(-\pi;\pi)$.
• Saks R. S. Cauchy problem for the Navier–Stokes equations, Fourier method. Pp. 53 - 79
Abstract: The Cauchy problem for the 3D Navier–Stokes equations with periodical conditions on the spatial variables is investigated. The vector functions under consideration are decomposed in Fourier series with respect to eigenfunctions of the curl operator. The problem is reduced to the Cauchy problem for Galerkin systems of ordinary differential equations with a simple structure. The program of reconstruction for these systems and numerical solutions of the Cauchy problems are realized. Several model problems are solved. The results are represented in a graphic form which illustrates the flows of the liquid. The linear homogeneous Cauchy problem is investigated in Gilbert spaces. Operator of this problem realizes isomorphism of these spaces. For a general case, some families of exact global solutions of the nonlinear Cauchy problem are found. Moreover, two Gilbert spaces with limited sequences of Galerkin approximations are written out.
• Startsev S. Ya. Necessary conditions of Darboux integrability for differential-difference equations of a special kind. Pp. 80 - 84
Abstract: This work dwells upon chains of differential equations of the form $\varphi(x,u_{i+1},(u_{i+1})_x)=\psi(x,u_i,(u_i)_x)$, where $u$ depends on the discrete variable $i$ and the continuous variable $x$, and the functions $\varphi(x,y,z)$, $\psi(x,y,z)$ and $x$ are functionally-independent. We demonstrate that necessary Darboux integrability conditions for chains of the above form can be easily derived from already known results. These conditions are not sufficient but may be useful for classification of Darboux-integrable differential-difference equations. As an auxiliary result, we also prove a proposition about structure of symmetries for differential-difference equations of a more general form.
• Timofeev A. Y. Construction of functions with determined behavior $T_G(b)(z)$ at a singular point. Pp. 85 - 93
Abstract: I. N. Vekua developed the theory of generalized analytic functions, i.e., solutions of the equation $$$\partial_{\overline z}w+A(z)w+B(z)\overline w=0, \tag{0.1}$$$ where $z\in G$ ($G$, for example, is the unit disk on a complex plane) and the coefficients $A(z)$, $B(z)$ belong to $L_p(G)$, $p>2$. The Vekua theory for the solutions of $(0.1)$ is closely related to the theory of holomorphic functions due to the so-called similarity principle. In this case, the $T_G$-operator plays an important role. The $T_G$-operator is right-inverse to $\frac\partial{\partial\overline z}$, where $\frac\partial{\partial\overline z}$ is understood in Sobolev's sense. The author suggests a scheme for constructing the function $b(z)$ in the unit disk $G$ with determined behavior $T_G(b)(z)$ at a singular point $z=0$, where $T_G$ is an integral Vekua operator. The paper states the conditions for $b(z)$ under which the function $T_G(b)(z)$ is continuous.