# Volume 52, № 2, 2000

### Criterion of unitary similarity of minimal passive scattering systems with a given transfer function

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 147-156

We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.

### On generalized local time for the process of brownian motion

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 157-164

We prove that the functionals \(\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,\) of a *d*-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_{Γ}(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂*R* ^{ d }, *k*≥1 and acting on finite continuous functions φ(·) in *R* ^{d} according to the rule \((\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),\) where ι(·) is a surface measure on Γ.

### Resolvent kernels that constitute an approximation of the identity and linear heat-transfer problems

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 165-182

Sufficient conditions are obtained for a Volterra integral equation whose kernel depends on an increasing parameter a to admit an approximation of the identity with respect to a in the form of a resolvent kernel. In this case, the solution of the integral equation tends to zero as a tends to infinity, and we establish estimates of this convergence in *L*. These results are used for obtaining estimates of the convergence of linear heat-transfer boundary conditions to Dirichlet ones as the heat-transfer coefficient tends to infinity.

### Approximation of classes of periodic functions with small smoothness

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 183-196

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces *C* and *L* by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (*n* - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.

### On the functional polystability of certain essentially nonlinear nonautonomous differential systems

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 197-207

For essentially nonlinear differential systems with the limit matrix of coefficients of the first-approximation system, we establish sufficient conditions for functional polystability, which generalizes the notion of exponential polystability.

### Ruin problem for an inhomogeneous semicontinuous integer-valued process

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 208-219

For a process ξ(*t* = ξ_{1}(*t*)+χ(*t*), *t*≥0, ξ(0) = 0, inhomogeneous with respect to time, we investigate the ruin problem associated with the corresponding random walk in a finite interval, (here, ξ_{1} (*t*) is a homogeneous Poisson process with positive integer-valued jumps and χ(*t*) is an inhomogeneous lower-semicontinuous process with integer-valued jumps ξ_{ n }≥-1).

### A generalization of the rogosinski-rogosinski theorem

Dekanov S. Ya., Mikhalin G. A.

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 220-227

We establish necessary and sufficient conditions for numerical functions α*j*(*x*), *j* ∈ *N*, *x* ∈ *X*, under which the conditions *K*(*f* _{ j } ⊂ *K*(*f* _{1}) ∀*j*≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) The functions *fj*(*x*) are uniformly bounded on the set *X* and take values in a boundedly compact space *L*, and *K(fj)* is the kernel of the function *fj*. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where *X* = *N*, α_{ j }(*x*) ≡ α_{j}, and the space *L* is the *m*-dimensional Euclidean space.

### Relationship between spectral and coefficient criteria of mean-square stability for systems of linear stochastic differential and difference equations

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 228-233

We establish the relationship (equivalence) between the spectral and algebraic (coefficient) criteria (the latter is represented in terms of the Sylvester matrix algebraic equation) of mean-square asymptotic stability for three classes of systems of linear equations with varying random perturbations of coefficients, namely, the ltô differential stochastic equations, difference stochastic equations with discrete time, and difference stochastic equations with continuous time.

### On asymptotically optimal weight quadrature formulas on classes of differentiable functions

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 234-248

We investigate the problem of asymptotically optimal quadrature formulas with continuous weight function on classes of differentiable functions.

### Convergence rates in regularization for the case of monotone perturbations

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 249-256

Convergence rates are justified for regularized solutions of a Hammerstein operator equation of the form *x* + *F* _{2} *F* _{1}(*x*) = *f* in the Banach space with monotone perturbations *f* _{2} ^{ h } and *f* _{1} ^{ h } .

### Distribution of values of additive functions on short intervals

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 257-265

We consider a family of completely additive functions β_{q}(*n*) defined on the set of natural numbers. We find an asymptotic expression for the summation function Σ_{ n≤x }β_{ q }(*n* study its distribution on short intervals.

### On the rosenthal inequality for mixing fields

Fazekas L., Kukush A. G., Tómács T.

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 266-276

A proof of the Rosenthal inequality for α-mixing random fields is given. The statements and proofs are modifications of the corresponding results obtained by Doukhan and Utev.

### On the endomorphisms of translation modules of polynomials

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 277-281

We determine the structure of a ring of endomorphisms of a translation module whose structure is determined by a group of translations of an affine space that acts by means of displacement on a polynomial algebra.

### On local perturbations of linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 282-287

We investigate the problem of preservation of regularity of linear extensions of dynamic systems on a torus under perturbations.

### The third ukrainian-scandinavian conference on probability theory and mathematical statistics

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 288