Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.      

Generalized moment representations and invariance properties of Padé approximants

By the method of generalized moment representations, we generalize the well-known invariance properties of Padé approximants under linear-fractional transformations of approximated functions.     

Solution of the unconditional extremum problem for a linear-fractional integral functional on a set of probability measures

A new, generalized and strengthened, form of an assertion about an extremum of a linear-fractional integral functional given on a set of probability measures is presented. It is shown that the solution of the extremal problem for such a functional is completely determined by the extremal properties of the so-called test function, which is the ratio of the integrands of the numerator and the denominator. On the basis of this assertion, a theorem on an optimal strategy for controlling a semi-Markov process with a finite set of states is proved. In particular, it is established that if the test function of the objective functional of a control problem attains a global extremum, then an optimal control strategy exists, is deterministic, and is determined by the point of global extremum. The corresponding assertions are also obtained for the case where the test function does not attain the global extremum.     

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefficient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89–138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287–317, 2002)     

Optimization Problem on Permutations with Linear-Fractional Objective Function: Properties of the Set of Admissible Solutions

We consider an optimization problem on permutations with a linear-fractional objective function. We investigate the properties of the domain of admissible solutions of the problem.     

General linear-fractional branching processes with discrete time

We study a linear-fractional Bienaymé–Galton–Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

Asymptotically optimal estimation in a linear-fractional regression problem with random errors in coefficients

Considering the linear-fractional regression problem with errors in independent variables, we construct and study asymptotically optimal estimators for unknown parameters in the case of violation of the classical regression assumptions (the variances of the observations are different and depend on the unknown parameters).     

Classification of Certain Class of Ordinary Differential Equations of the First Order

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.     

On an approach to the modelling of problems connected with conflicting economic interests

The aim of the present paper is to show a possible way to use the recent results obtained in linear and linear-fractional programming in applications connected with the coordination of conflicting economic interests. We consider the relationship between linear and linear-fractional programming problems with the same feasible set and show how it is possible to reorientate objective functions such that all objective functions considered lead to the same optimal solution. We show also that in certain cases the dual estimates of linear and linear-fractional programming are closely connected. The economic interpretation of the results obtained is outlined and a numerical example is presented.     

Operator linear-fractional relations: main properties, some applications

We consider operator linear-fractional relations of the form $$ F(K)=\left\{ {Q:A+BK=Q\left( {C+DK} \right)} \right\}, $$ where A, B, C, D, K, and Q are operators between Hilbert spaces. If C + DK is invertible, the relation F becomes a linear-fractional transformation. In the case where F is the automorphism of a unit operator ball, we study the conditions for F to be represented in the form of a composition of an automorphism and an affine relation. The results obtained are applied to the Abel–Schröder equations, Königs embedding problem, and some other questions.     

The Fredholm Index for Elements of Toeplitz-Composition C<Superscript>*</Superscript>-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C^*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C^*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.     

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H² by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H² that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.     

Basis of absolute invariants of completely solvable linear and linear-fractional discrete dynamical systems

We construct a basis of nondegenerate absolute invariants of completely solvable linear and linear-fractional discrete dynamical systems.     

Linear-fractional branching processes with countably many types

We study multi-type Bienaymé–Galton–Watson processes with linear-fractional reproduction laws using various analytical tools like the contour process, spinal representation, Perron–Frobenius theorem for countable matrices, and renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for R$R$-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.     

On Lacunary Analogs of the Poincare Theta-Series and Their Applications

We consider Fuchsian groups of linear-fractional transformations such that each vertex of the fundamental polygon is common for an even or infinite number of fundamental congruent polygons meeting at this point. The whole collection of transformations splits into two disjoint sets. For these sets we introduce two lacunary kernels whose sum represents the well-known analog of Chibrikova and Sil'vestrov's kernel and study their properties. We introduce automorphic forms of dimension –4m that differ from the Poincare theta-series. We indicate an application of one of the constructed lacunary kernels which does not include the Cauchy kernel to solving some boundary value problem with a shift of the contour inside the domain.     

Some properties of a linear-fractional transformation

Seven properties of a linear-fractional analytic function, many of which are also valid in the domain of real variables, are pointed out. In either case, these properties are important for applications to problems of subterranean hydromechanics.     

On linear-fractional relations and images of angular operators

For plus-operators in a Banach indefinite space, we consider a linear-fractional relation. The classes of operators with the empty domain of definition for such a relation are described. The sufficient (and necessary, in some meaning) conditions for the chain rule to be valid are given.     

Constructing an automorphic form from the orbit of a transformation

We consider the Fuchsian groups of linear-fractional transformations. We propose a new method for presenting automorphic forms as gap series over an appropriate subset of transformations of the group which is not a subgroup. Comparative analysis of the Poincaré theta-series and gap series demonstrates that the use of gap series requires less transformations and parameters that the summands of series depend on.     

关于二元分式线性齐次复合函数的积分

通过把二重积分转化成曲线积分,得到二元分式线性齐次复合函数积分的两个定理及其若干推论,并构造一些比较典型的用一般方法难以计算的例子说明结论的应用.