# Volume 46, № 3, 1994

### Mark Grigor'evich Krein

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 146

### Unitary colligations and parametrization formulas

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 147–154

The aim of this paper is to describe, in a rather concise but relatively self-contained manner, some relations between unitary colligations which are regarded as linear systems, and parametrization formulas for the solutions of some interpolation problems.

### Spectral theory of a string

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 155–176

In this survey, we present the principal results of Krein's spectral theory of a string and describe its development by other authors.

### Spectral approach to white noise analysis

Berezansky Yu. M., Livinskii V. O., Lytvynov E. V.

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 177–197

By using the spectral projection theorem, we construct the classical Segal transformation as a Fourier transformation in the generalized joint eigenvectors of a certain family of field operators. It is noted that the spectral approach to the Segal transformation, which forms the basis of the analysis of Gaussian white noise, enables one to construct a significant generalization of this transformation.

### On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 198–211

This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic analysis on topological groups. These papers are known to be classical and form the basis of numerous contemporary researches into these fields.

### Uniqueness theorems for rational, algebraic, and algebroid functions

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 212–226

A number of points $A$, for which one must define the sets of simple $A$-points to determine a rational function, a polynomial, and an algebraic or algebroid function uniquely, is obtained.

### A class of admissible functions in the principle of invariance of wave operators

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 227–229

A class of admissible functions that satisfies the well-known invariance principle in the scattering theory is extended.

### A relation between two results about entire functions of exponential type

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 230–243

It is shown that the Beurling - Maliiavin multiplier theorem for the entire functions of exponential type can be derived from certain estimates for polynomials on the complex plane.

### Estimation of the solutions of the Sturm-Liouville equation

Levin B. Ya., Mirochnik L. Ya.

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 244–278

Exact estimates are presented for the solutions of the problem $\ddot y + \lambda ^2 p(t)y = 0, y(0) = 0, \dot y(0) = 1$ with $p(t)$ satisfying one of the following conditions: $$(i) |p(t)| \leqslant M< \infty ; (ii) 0< \omega _1 \leqslant p(t) \leqslant \omega _2< \infty ; (iii) \mathop {sup}\limits_x \int_x^{x + T} {p(t)dt = P_T /T.}$$ The extremal solutions are found.

### Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 279–292

For the equation $L_0x(t)+L_1x′(t) + ... + L_nx^{(n)}(t) = O$, where $L_k, k = 0,1,...,n$, are operators acting in a Banach space, we establish criteria for an arbitrary solution $x(t)$ to be zero provided that the following conditions are satisfied: $x^{(1−1)} (a) = 0, 1 = 1, ..., p$, and $x^{(1−1)} (b) = 0, 1 = 1,...,q$, for $-∞ < a < b < ∞$ (the case of a finite segment) or $x^{(1−1)} (a) = 0, 1 = 1,...,p,$ under the assumption that a solution $x(t)$ is summable on the semiaxis $t ≥ a$ with its first $n$ derivatives.

### Factorization of operators. Theory and applications

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 293–304

A survey of the development of Krein's factorization method and its applications is given.

### Mark Grigor'evich Krein (recollections)

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 305-309

### Mark Grigor'evich Krein in Kuibyshev (memoirs of a post-graduate student)

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 310-311