# Volume 57, № 5, 2005

### Yurij Makarovich Berezansky (the 80th anniversary of his birth)

Gorbachuk M. L., Gorbachuk V. I., Kondratiev Yu. G., Kostyuchenko A. G., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11

### Functional Analysis in the Institute of Mathematics of the National Academy of Sciences of Ukraine

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 582–600

We give a brief survey of results on functional analysis obtained at the Institute of Mathematics of the Ukrainian National Academy of Sciences from the day of its foundation.

### On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 601–613

We study representations of solutions of the Dirac equation, properties of spectral data, and inverse problems for the Dirac operator on a finite interval with discontinuity conditions inside the interval.

### Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials

Atakishiyev N. M., Klimyk A. U.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 614–621

By using two operators representable by Jacobi matrices, we introduce a family of *q*-orthogonal polynomials, which turn out to be dual with respect to alternative *q*-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.

### Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces

Bozhok R. V., Koshmanenko V. D.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 622–632

Let *A* be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of *A*). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + = M \oplus N_ +\) so that the subspace \(M_ +\) is dense in \(H_0\). We construct and study a singularly perturbed operator *A* associated with a new rigging \(H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +\), where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )\), and establish the relationship between the operators *A* and *A*.

### Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

Gorbachuk M. L., Hrushka Ya. I., Torba S. M.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 633–643

For an arbitrary self-adjoint operator *B* in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator *B*, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator *B*, and the *k*-modulus of continuity of the vector *x* with respect to the operator *B*. The results are used for finding *a priori* estimates for the Ritz approximate solutions of operator equations in a Hilbert space.

### Conditional Expectations on Compact Quantum Groups and Quantum Double Cosets

Chapovsky Yu., Kalyuzhnyi A. A., Podkolzin G. B.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 644–653

We prove that a conditional expectation on a compact quantum group that satisfies certain conditions can be decomposed into a composition of two conditional expectations one of which is associated with quantum double cosets and the other preserves the counit.

### Point Spectrum of Singularly Perturbed Self-Adjoint Operators

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 654–658

We study the inverse spectral problem for the point spectrum of singularly perturbed self-adjoint operators.

### Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 659–668

We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators.

### Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 669–678

In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb{F}_q$ -linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u \circ u ... \circ u$ of the unknown function. As an algebraic background, imbeddings of the composition ring of $\mathbb{F}_q$ -linear holomorphic functions into skew fields are considered.

### On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 679–688

For a singular perturbation $A = A_0 + \sum^n_{i, j=1}t_{ij} \langle \psi_j, \cdot \rangle \psi_i,\quad n \leq \infty$ of a positive self-adjoint operator $A_0$ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations $A_T$ is carried out and the scattering matrix $\mathfrak{S}_{(A_T, A_0)}(\delta)$ is calculated in terms of parameters $t_{ij}$ under some additional restrictions on singular elements $\psi_{j}$ that provides the possibility of application of the Lax -Phillips approach in the scattering theory.

### Elliptic Operators in a Refined Scale of Functional Spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 689–696

We study the theory of elliptic boundary-value problems in the refined two-sided scale of the Hormander spaces $H^{s, \varphi}$, where $s \in R,\quad \varphi$ is a functional parameter slowly varying on $+\infty$. In the case of the Sobolev spaces $H^{s}$, the function $\varphi(|\xi|) \equiv 1$. We establish that the considered operators possess the properties of the Fredholm operators, and the solutions are globally and locally regular.

### On the Group $C^{*}$-Algebras of a Semidirect Product of Commutative and Finite Groups

Samoilenko Yu. S., Yushchenko K. Yu.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 697–705

By using representations of general position and their properties, we give the description of group $C^{*}$-algebras for semidirect products $\mathbb{Z}^d \times G_f$, where $G_f$ is a finite group, in terms of algebras of continuous matrix-functions defined on some compact set with boundary conditions.
We present examples of the $C^{*}$-algebras of affine Coxeter groups.

### Multifractal Analysis of Singularly Continuous Probability Measures

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 706–720

We analyze correlations between different approaches to the definition of the Hausdorff dimension of singular probability measures on the basis of fractal analysis of essential supports of these measures. We introduce characteristic multifractal measures of the first and higher orders. Using these measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.