L. A. Sakhnovich

Interpolation Theory and Its Applications

Introduction. 1. Operator Identities and Interpolation Problems. 2. Interpolation Problems in the Unite Circle. 3. Hermitian-Positive Functions of Several Variables. 4. De Branges Spaces of Entire Functions. 5. Degenerate Problems (Matrix Case). 6. Concrete Interpolation Problems. 7. Extremal Problems. 8. Spectral Problems for Canonical Systems of Difference Equations. 9. Integrable Nonlinear Equations (Discrete Case). 10. On Semi-Infinite Toda Chain. 11. Functions with an Operator Argument. Commentaries and Remarks. Bibliography. Index.

Integral equations with difference kernels on finite intervals

1. An Invertible Operator with a Difference Kernel.- 1. Constructing the inverse operator.- 2. Existence conditions and the structure of the inverse operator.- 3. Equations with a special right-hand side.- 4. Operators with a difference kernel in the space Lp(0,?).- 5. The use of the Fourier transform.- 6. Toeplitz matrices.- 2. Equations of the first kind with a Difference Kernel.- 1. Equations of the first kind with a special right-hand side.- 2. Solutions of equations of the first kind.- 3. Generalized solutions.- 4. On the behavior of solutions.- 5. On one class of integro-differential equations.- 3. Examples and Applications.- 1. Integral equations with kernels of power type.- 2. Integral equations with logarithmic type kernels.- 3. Regularization.- 4. Fractional integrals of purely imaginary order.- 5. On a class of integral equations which are solvable in exact form.- 6. On certain problems of hydrodynamics.- 7. Equations on the contact theory of elasticity.- 8. The equation of radiation transfer.- 4. Eigensubspaces and Fourier transform.- 1. Classification of eigensubspaces.- 2. On the distribution of the roots of Fourier images.- 5. Operator Bezoutiant and Roots of Entire Functions.- 1. Definition and properties of the operator B.- 2. Operator T corresponding to a pair of entire functions.- 6. Operator Identities and Systems of Equations with W-difference Kernel.- 1. The principal notions of S-knot theory.- 2. Systems with W -difference kernels.- 3. Prandtl equation.- 7. Integral Equations in the Theory of Stable Processes.- 1. The deduction of the integro-differential equations.- 2. Solution of the Kac problems.- 3. Two-sided estimation of the smallest eigenvalue of the operator.- 8. Problems of Communication Theory.- 1. Problem of optimal prediction.- 2. Problem of diffraction on a strip.- 3. Extremal problems in the theory of synthesis of antennae.- Commentaries and Remarks.- References.

Half-inverse problems on the finite interval

The spectrum of the Sturm-Liouville problem is given. The potential on the half-interval is known. It is necessary to reconstruct the potential on the whole interval. The uniqueness of the solution of the formulated problem is proved (see Hochstadt H and Lieberman B 1978 SIAM J. Appl. Math. 34 676-80, Hald O 1984 Commun. Pure Appl. Math. 37 539-77, Gesztesy F and Simon B 2000 Trans. AMS 352 2765-89). Under some conditions in this paper we prove the existence of the solution and give a method of reconstructing this solution.

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapovs method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.

An Operator Approach to the Potapov Scheme for the Solution of Interpolation Problems

Interpolation problems play a significant role both in applied and theoretical investigations. Classical interpolation problems include those of Nevanlinna-Pick, Caratheodory, Schur, and Hamburger, as well as the problem of trigonometric moments.

Spectral theory of a class of canonical differential systems

The general spectral theory of canonical systems is used to study a generalized Krein system. Direct and inverse problems for this system are considered. In particular, some proofs are supplied for Kreins results published by him without proof.

Inverse Scattering Problem for Continuous Transmission Lines with Rational Reflection Coefficient Function

In this paper we obtain explicit formula for the reflexivity coefficient function (or potential) of an ordinary differential operator if its reflection coefficient is a rational matrix valued function. The solution is given in terms of a realization of the reflection coefficient function.

Some indefinite cases of spectral problems for canonical systems of difference equations

An approach by operator identities is used to investigate some direct and inverse problems of spectral theory for canonical systems of difference equations in the indefinite case.

Inverse problems for equations systems

In inverse problems it is necessary to restore a system from some information about this system (scattering data, spectral data). In the present case the following problems have to be solved: n nI. n nTo prove that there exists a system corresponding to the data (existence theorem). n n n n nII. n nTo prove that there is a unique system corresponding to the data. n n n n nIII. n nTo find the method of constructing this system by the data.

Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

To introduce the main notions of triangular factorization consider a Hilbert space ( L^2(a,b),,,,,( infty leq a < b leq infty) )The orthogonal projectors ( P_ xi) in (L^2(a,b)) are defined by the relations ( (P_xi f)(x)= f(x) for , a < x < xi, ,, (P_xi f)(x)= 0 for xi < x < b (f epsilon L^2(a,b))) Denote the identity operator by I.