The class $H_{\rho_1, \rho_2, \ldots,\rho_n}$ ofholomorphic functions $f(z), z \in \C,$ is consideredsuch that each function $f(z)$ can be represented by the following way$$f(z) = \Delta_1(\arg z) |z|^{\rho_1} + \Delta_2(\arg z) |z|^{\rho_2}+ \ldots + \Delta_n(\arg z) |z|^{\rho_n} + \kappa(z),$$$$\ z \to \infty,$$where $ 0 < [\rho_1] < \rho_n < \ldots < \rho_1$, andthe function $\kappa(z)$ satisfies$$\int\limits_T^{2T} \sup \limits_{\arg z \in [0, 2 \pi]} |\kappa(z)|^q d |z| =o (T^{\rho_n q + 1}), \ \ T \to \infty, \ q > 1.1991 Mathematics Subject Classification 30E25. Top of the page.
Convolution equations on complex hyperbolic space $H^n(\C)$ are studied.The general solution of one class of such equations has been found.The local two-radii theorem for system convolution equationshas been obtained.1991 Mathematics Subject Classification 53C65. Top of the page.
One of the most important problems in the theory of subharmonicfunctions in angular domains is the problem of connection between the growthof the subharmonic functions and the distribution of associated measures.Wegive a compact presentation of general properties of subharmonic functionsin a complex half-plane, based on the concept of completemeasure. This is a new version of some previous resultsof one of the authors.1991 Mathematics Subject Classification 31A05, 31A10. Top of the page.
The angular boundary value is described for theclass of analytic functions in the half-plane$\Re z>0$, for which\begin{equation}\sup\limits_{|\varphi|<\frac{\pi}{2}}\left\{\int\limits_{0}^{+\infty}|f(re^{i\varphi})|e^{-\sigma r|\sin\varphi |}dr\right\}<+\infty,\qquad 0\leq\sigma<+\infty.\end{equation}1991 Mathematics Subject Classification 30E05. Top of the page.
Let $\{\hat P_n\}$ be a system of polynomials orthonormal on thecompact subset $K$ of the complex plane with relating to a measure $d\mu$.The logarithmic asymptotics for these polynomials are characterized in the termsof supremum and $L^2(d\mu)$ norms inequalities.1991 Mathematics Subject Classification 42C05. Top of the page.
Under fairly general condition on measure, some unsharpened asymptoticformulae for its Newton potential are found.1991 Mathematics Subject Classification 30B50. Top of the page.
A new necessary condition for a set to be the zero set of an absolutely monotonic function is given. If $A \subset \{z: \mbox{Re}~z <0\}$ is the zero set of an absolutely monotonic function then for any $\beta \in (0,\pi/2)$ there exists a nonnegative continuous function $h_\beta$, $ h_\beta \in L^1 (-\infty, -1]$, such that $$\sum _{a\in A \bigcap \{a: |\arg a-\pi |< \beta\} } \frac {1}{(x-a )^2 } \leq h_\beta (x).$$ It is shown that this condition is not a consequence of conditions known before. 1991 Mathematics Subject Classification 30D50, 44A10. Top of the page.
The upper estimate of $p$-strong means ($1 \le p \le 2$) by disks forFourier series of almost everywhere bounded functions defined on thetwo-dimensional torus is obtained. Analog of well-known one-dimen- sionalFomin's theorem is proved for trigonometric series of a special form. 1991 Mathematics Subject Classification 42B08. Top of the page.
Upper estimates of the rate of the uniform approximation byharmonic polynomials are estabblished for harmonic functions on compactsets in $\R^3$ John's areas added. These results continuethe Andriyevskii's investigations.1991 Mathematics Subject Classification 41A25. Top of the page.
Summary. This article is a review of main results from \S\S \ 5, 6of the author's preprint (1998).Analytical proximate orders generated by entire or meromorphicfunctions are studied.Asymptotics of associated with them "gamma-functions" and"exponents" , i.e. analogs of Mittag-Lefflerfunctions, are investigated. The generalized indicator ofthe "exponent" of order $\rho\in(1/2,1)$ and of order$1$ and minimal type was shown to be trigonometric on theinterval $[-\pi,\pi]$.It is demonstrated two applications of these results.1991 Mathematics Subject Classification 30D20. Top of the page.
Let $\gamma(r)$ -- be a growth function that is a stricly positive continiousuncreasing unbounded function on the ray $[0,\infty)$. We introduce the classof just subharmonie in the half-plane functions of finite $\gamma$-type andgive criterions for functions belonging to the class by the Fourier coefficients.1991 Mathematics Subject Classification 31A10. Top of the page.
For meromorphic functions in the circle we study the values $\hat b(a,f)$ similiar those introduced by A.~Eremenko. Sharp estimates and a defect relation are obtained.1991 Mathematics Subject Classification 30D35. Top of the page.
A new sharpness of classical Morera theorem in the unit diskhas been obtained. We consider integrals over boundaries ofcongruent sectors.1991 Mathematics Subject Classification 53C65. Top of the page.
The problem of describing the range of a Bergman space $B_2(G)$ under the Cauchy transform $K$ for a Jordan domain $G$ was solved by Napalkov (Jr) and Yulmukhametov. It turned out that $K(B_2(G))=B_2^1(C\overline{G})$ if and only if $G$ is a quasidisk; here $B_2^1(C\overline{G})$ is a Dirichlet space of the complement of $\overline{G}$. The description of $K(B_2(G))$ for an integrable Jordan domain is given by the author earlier. In the present paper we give a description of $K(B_2(G,\omega))$ for a weighted Bergman space $B_2(G,\omega)$ with a weight $\omega$ which is constant on level lines of the Green function of $G$. In case $G={\Bbb D}$, the unit disk, and under some additional conditions on the weight $\omega$, $K(B_2({\Bbb D},\omega))=B_2^1(C\overline{\Bbb{D}}, \omega^{-1})$, a weighted analogy of a Dirichlet space.1991 Mathematics Subject Classification 30H05. Top of the page.
We get an estimate Benedics-Sodin type for the harmonic measure of a set in the qube1991 Mathematics Subject Classification 31‚05, 31‚25. Top of the page.
Let $E$ be the class of infinite dimensional semisimple Banach algebraswith unit elements provided that their subalgebras generated by an arbitrary nonscalar element and the unit areisometrically isomorphic to the algebras themselves. In this paper, thestructure of such algebras in $E$ will be studied. It is shown that the disc algebra is in $E$ and that the interiorsof the maximal ideal spaces of algebras in $E$ are not empty. Moreover, if an algebra in $E$ is uniform then it is possible to give an analytical structure in itsmaximal ideal space.1991 Mathematics Subject Classification 46J10, Secondary 30C. Top of the page.
The solutions of some combined problems of theory of elasticity withsummetrical axis for cone may be found by their reduction to the boundaryRiemannian problem for analytical functions. Factorization of the coefficientof the Riemannian problem, wich is the quotient of the linear functions toLegendre functions, is done in canonical products. The asymptotic formularfor zeros and poles of the coefficient of the Riemannian problem are obtained,with the help of which the asymptotic estimations of the canonical productsare found.1991 Mathematics Subject Classification 73C02. Top of the page.
Relations are studied between directional and partialLelong numbers and residual Monge-Amp\`ere measures forplurisubharmonic functions with multicircled singularities.1991 Mathematics Subject Classification 32F05. Top of the page.
The boundary problem of the theory of rigid wedge with elastic layer isstudied. The obtained integral equation is solved numerically.The distribution of the contact tension is found.1991 Mathematics Subject Classification 73C02. Top of the page.
We investigate the uniform approximation of analytic functions on compact sets by polynomials with special restrictions for their coefficients. The coefficients are assumed to lie on a ray or two rays, to be positive, integer or natural. The order of approximation is estimated as well.1991 Mathematics Subject Classification 41A60. Top of the page.
In this work the problem of stabilizability by feedback control isinvestigated for evolution systems of partial differential equations on$\mbox{\bbb R}^n\times[0,+\infty)$. We obtain a criterion (a necessary andsufficient condition) for stabilizability of these systems. To prove thecriterion, we use the Fourier transform method. To apply this method, wehave to investigate deformations of algebraic hypersurfaces under somepolynomial perturbations and have to obtain estimates of semi-algebraicfunctions on semi-algebraic sets by using of the Tarski---Seidenberg theoremand its corollaries. In the work we also analyse the obtained criterion andgive some examples of stabilizable and nonstabilizable systems. Top of the page.
We present the first fragements of a functionalanalytic treatment of classical Dirichlet series. The problemscenter around the Dirichlet series version of the Hardy space.We mention what the space of multipliers is, as well as thecollection of bounded composition operators. Convergence issuesare also mentioned. The focus in the paper is on open problems.1991 Mathematics Subject Classification 30B50. Top of the page.
We give new Criterions of resolvibility of interpolation problem is classesof entire functions with special conditions for growth. We studythe case of analytic functions in the disc as well.1991 Mathematics Subject Classification 30E05. Top of the page.
Classes of analytic functions in the unit disk with bounded derivatives of general type in the spaces $ H_\infty $ and $ H_1 $ are considered. Problems of approximation of such functions by partial Taylor sums are studied. Top of the page.
In this work a generalized Nevanlinna-Pick problem in a special subclassof Nevanlinna functions are considered. Explicit formulas for theextremal solutions of this problem are obtained. Top of the page.
Two mixed boundary value problems for thermoelastic media are underconsideration. Their solutions are represented by the dynamic analogues ofthermoelastic single and double-layer potentials. These representations leadto systems of nonstationary boundary equations. The unique solvability ofthese systems is proved in one-parameter scale of Sobolev type functionspaces. Top of the page.
The regularization of a diffraction problem of field of the grate ofvertical magnetic dipoles on a coating ball is carried out with help of the inversion of anintegral equation of Abel's type and the partial inversion of dual series equations. The system of algebraic equations of II kind with a compactoperator in $\ell^2$ is obtained. The system has high numerical stability. Top of the page.
The basic dynamic problems for thin elastic plates in Kirchhoffmodel are under consideration. The representation of the solutions to theproblems by the dynamic analogs of the single- and double layer potentialswith respect to their unknown densities. The solvability ofthese systems is proved in one-parameter scale of Sobolev-type functionspaces. Top of the page.
In this article a criterion for well-posedness of a boundary-value problemin a bilayer is obtained and existence of well posed boundary-value problemis proved for any system of pseudo-differential equations.It is also studied for which systems parabolic boundary-value problems exist. Top of the page.
The paper deals with the problem about an embedding of the group $S_f$of finite permutations of a countable set into a locally compact group. Ashas been proved in the paper, the group $S_f$ cannot be embeded into an almost connected locally compact group and can be embeded into a continuos totally disconnected locally compact group tightly. Top of the page.
In the work the solution of integral equations with a matrix kernel problem is studied. A method is given to construct inverse oparators. This method is based on using operation equalities. Top of the page.
In the paper we study totally geodesic submanifods of surfaces with therevolution group $SO(2)\times SO(2)$ in Euclidean spaceand geodesic lines on surfaces with revolution groups of the form$SO(n_1)\times \dots\times SO(n_k)$. Top of the page.
This article is devoted to studying of the Grassmannian image ofthe surfaces with flat normal connection. The theorem is proved: If $F^l$ isthe surface with zero sectional curvature, flat normal connection and with constant index of nullity in Euclidean space $E^{l+p}$, then Grassmannianimage $ƒ(F^l)$ is supplied with flat metrics.In case of nondegenerate Grassmannian image this theorem is proved by Muto Y.1991 Mathematics Subject Classification 42A70. Top of the page.
In work is shown that for completely control system of a type$$ \dot x=Cx+u, \quad u\in{\bf K}, $$ where $K$ -- is a convex closed cone,problem of stabilization (problem of construction regulator) can be solved with any forward given degree of stability, but, consequently, and for control system $$\dot x=f(x,u), \quad u\in{\bf U}\subset {\bf R}^r, $$ where exist $u_0\in U:f(0,u_0)=0$, find enough small neighbourhood, from any point which problem of construction regulatorsolvable with any forward given degree of stability. Top of the page.
The existence of non-trivial deformations of a surface $F^2$ in $E^4$ preserving its Grassmann image and Gauss curvature is studied.It is proved that a surface $F^2$ in $E^4$ allows non-trivial deformations preserving its parabolic Grassmann image and Gauss curvatureif and only if it is ruled. Top of the page.
The complete controllability is proved for a class of triangular systems of ordinary differential equations whose trajectories, in general, can not be mapped by a locally diffeomorphical at every point change of variables onto the trajectories of the canonical systems. The considered class of trangular systems contains as a special case the triangular systems studied before. One example is considered. Top of the page.
Special solutions for shock wave dynamics equations in the form of simple waves are investigated.The obtained self-similar solution for cylindrical shock waves describesconvergent shock wave spreading with the intensity varying along the frontwithin the narrowing spiral cavity. This solution is not axially symmetric onebut a cumulative ef\/fect in this case is the same as in the axially symmetriccase. Top of the page.
In a Hilbert space we consider the differential equation$\sum\limits_{j=0}^{n}A_ju^{(j)}(t)=0$with linear closed operators. The operators $A_j$can have, in general, non-trivial kernels. We laydown the conditions under which solutions of this equationsallow an asymptotic expansion to aseries in elementary solutions when $t\to \infty$.The results are applied to partial differential equations. Top of the page.
In the given operation through the spectral theory of systems ofcommuting operators the spectral expansions vectorial evolutioneryof representable fields are obtained. The spectral representationssystems of commuting dissipative operators are constructed inpurticular. Top of the page.
The problem of synthesis of positional control for controllableprocess discribed by a linear differential equation with unboundedoperator in Hilbert space is considered. A restricted positional controlsuch that the solution of corresponding equation with any initial conditiontends to zero for a finite time is constructed. The time is equal to the value of cotrollability functional at initial state.The examples of controllable processis for which the control solves the considered problem is given. Top of the page.
A new approach to the forming of optimal mean square filters, based on the theoryof operator colligations has been realized. Solving of Wiener-Hopf equationhas beeb found in an explicit form. Top of the page.
The correlations theory of the nonstationary sequences in Hilbert Spacedetermined by twice commutative operators has been discussed in thispaper. The characterictics connected with the nonselfadjoint of thecorresponding operators have been introduced. The triangle Zolotariyov'smodels of twice commutative operators have been used for studing suchsequences. Top of the page.
In the present paper the solution of the Dirichlet problemis represent as the limit of the solutions of the Neumannproblems for the same equation in the same domain. We usethe methodto solve the boundary problems for biharmonicequation proposed in ''Numarical analisys of the variationalinequalities" by R.G.~Glovinsky, J.-L. Lions, R Tremoulier. Top of the page.
In this paper the best valuation in mean square sense based on the meaning ofthe Wiener field on the two non-increase curves is constructed for meaning ofthe Wiener field in some point on the plane and its error is calculated. Top of the page.
For the rate of convergence of the solution of initial boundary-value problem for the parabolic equation with quick random oscillations to the solution of Ito equation in a uniform metric of probability, estimates are obtained. Top of the page.
The invariants of the extended Poincare algebras, have been used forredution of the two-dimensional and three-dimensional Born-Infeld equation. Top of the page.