Lev A. Sakhnovich

Inverse problems and nonlinear evolution equations : solutions, Darboux matrices and Weyl-Titchmarsh functions

This monograph fits the clearly need for books with a rigorous treatment of the inverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authors develop a unified treatment of explicit and global solutions via the transfer matrix function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained and starts with preliminaries and examples, and hence also serves as an introduction for advanced graduate students in the field.

ON A MEAN VALUE THEOREM IN THE CLASS OF HERGLOTZ FUNCTIONS AND ITS APPLICATIONS

Linear-fractional transformations of the pairs with J-property are considered. Ex- tremal functions from an important subclass obtained in this way are expressed as mean values of extremal functions from another subclass of these linear-fractional transformations. Applications to some spectral and interpolation problems are discussed.

Inverse Problems for Canonical Differential Equations with Singularities

The inverse problem for canonical differential equations is investigated for Hamiltonians with singularities. The usual notion of a spectral function is not adequate in this generality, and it is replaced by a more general notion of spectral data. The method of operator identities is used to describe a solution of the inverse problem in this setting. The solution is explicitly computable in many cases, and a number of examples are constructed.

Matched filtering for generalized stationary processes

The methods for solving optimal filtering problems in the case of the classical stationary processes have been well known since the late 1940s. Practice often gives rise to what is not a classical stationary process but a generalized one, and white noise is one simple example. Hence, it is of interest to describe the system action on the generalized stationary processes, and then to carry over filtering methods to them. For arbitrary generalized stochastic processes this seems to be a challenging problem. In this correspondence, we identify a rather general class of S/sub J/-generalized stationary processes for which the desired extension can be done for matched filters. This class can be considered as a model of colored noise, and it is wide enough to include white noise, positive frequencies white noise, as well as certain generalized processes occurring in practice, namely, when the smoothing effect gives rise to the situation in which the distribution of probabilities may not exist at some time instances. One advantage of the suggested model is that it connects optimal filter design with inverting of integral operators; the methods for the latter can be found in the extensive literature.

ON THE KRE˘ IN-LANGER INTEGRAL REPRESENTATION OF GENERALIZED NEVANLINNA FUNCTIONS ∗

The Kre˘ in-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in terms of their integral representations.

On Indefinite Cases of Operator Identities Which Arise in Interpolation Theory. II

This paper studies operator identities from interpolation theory that are related to the generalized Caratheodory class of matrix-valued functions on the interior and exterior of the unit circle. A key tool is a Kreĭn–Langer integral representation for generalized Caratheodory functions that generalizes the Herglotz representation in the classical case. Every generalized Caratheodory function meeting certain conditions induces an operator identity. The main results of the paper characterize the class of identities which arise in this way. An application to a tangential interpolation problem is given. Parallel results for generalized Nevanlinna functions were obtained by the authors in a previous work.

Optimal Prediction of Generalized Stationary Processes

Methods for solving optimal filtering and prediction problems for the classical stationary processes are well known since the late forties. Practice often gives rise to what is called generalized stationary processes [GV61], e.g., to white noise and to many other examples. Hence it is of interest to carry over optimal prediction and filtering methods to them. For arbitrary generalized stochastic processes this could be a challenging problem. It was shown recently [OS04] that the generalized matched filtering problem can be efficiently solved for a rather general class of SJ-generalized stationary processes introduced in [S96]. Here it is observed that the optimal prediction problem admits an efficient solution for a slightly narrower class of TJ-generalized stationary processes. Examples indicate that the latter class is wide enough to include white noise, positive frequencies white noise, as well as generalized processes occurring when the smoothing effect gives rise to a situation in which the distribution of probabilities may not exist at some time instances. One advantage of the suggested approach is that it connects solving the optimal prediction problem with inverting the corresponding integral operators SJ. The methods for the latter, e.g., those using the Gohberg-Semencul formula, can be found in the extensive literature, and we include an illustrative example where a computationally efficient solution is feasible.

Nonlinear Fokker–Planck Equation: Stability, Distance and the Corresponding Extremal Problem in the Spatially Inhomogeneous Case

We start with a global Maxwellian M k , which is a stationary solution, with the constant total density \((\rho(t) \equiv {\tilde{\rho}})\), of the Fokker–Planck equation. The notion of distance between the function M k and an arbitrary solution f (with the same total density \({\tilde{\rho}}\) at the fixed moment t) of the Fokker–Planck equation is introduced. In this way, we essentially generalize the important Kullback–Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.