# Volume 62, № 8, 2010

### Generalized solutions for linear operators with weakened a priori inequalities

Anikushyn A. V., Nomirovs'kii D. A.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1011–1021

We propose an approach to the investigation of generalized solutions of linear operators that satisfy weakened a priori inequalities. This approach generalizes several well-known definitions of generalized solutions of operator equations. We prove existence and uniqueness theorems for a generalized solution.

### Completeness of invariant ideals in groups

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1022–1031

We introduce and study various notions of completeness of translation-invariant ideals in groups.

### Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes

Doronin V. G., Vakarchuk S. B.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1032–1043

We obtain exact Jackson-type inequalities in the case of the best mean square approximation by entire functions of finite degree $≤ σ$ on a straight line. For classes of functions defined via majorants of averaged smoothness characteristics $Ω_1(f, t ),\; t > 0$, we determine the exact values of the Kolmogorov mean ν-width, linear mean ν-width, and Bernstein mean $ν$-width, $ν > 0$.

### Regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ and ∗-algebra associatedwith it

Kruhlyak S. A., Livins'kyi I. V.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1044–1062

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets $\{0,1,2,3,4,5\}, \{0,2,4\}$, and $\{0,3\}$, respectively, and the equality $A + B + C = 6I$ is true.

### Localization of eigenvalues of polynomial matrices

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1063–1077

We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities. We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a system of linear matrix inequalities.

### Constructive description of monogenic functions in a harmonic algebra of the third rank

Plaksa S. A., Shpakovskii V. S.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1078–1091

By using analytic functions of a complex variable, we give a constructive description of monogenic functions that take values in a commutative harmonic algebra of the third rank over the field of complex numbers. We establish an isomorphism between algebras of monogenic functions in the case of transition from one harmonic basis to another.

### Examples of $C^1$-smoothly conjugate diffeomorphisms of the circle with break that are not $C^{1+γ}$ -smoothly conjugate

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1092–1105

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not $C^{1+γ}$-smoothly conjugate for any $γ > 0$. In this way, we show that the previous result concerning the $C^1$-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.

### Modules of continuity and analytic functions

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1106–1113

For a function analytic in a compact domain and continuous in its closure, it is shown that the modules of continuity on the boundary of the domain and in its closure coincide.

### Decomposability of matrix polynomials with commuting coefficients into a product of linear factors

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1114–1123

The list of known sets of factorizable matrix polynomials is supplemented by new sets of polynomials of this sort. The known set of nonfactorizable matrix polynomials is extended. These results can be applied to the study of polynomial equations and systems of differential equations with constant coefficients.

### Approximation of the classes $S^r_{p,θ}B(\mathbb{R}^d)$ of functions of many variables by entire functions of a special form

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1124–1138

Exact-order estimates are obtained for the approximations of the functional classes $S^r_{p,θ}B(\mathbb{R}^d)$ by entire functions of a special form.

### On a criterion of rationality for a series in orthogonal polynomials

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1139–1144

We present a criterion of rationality for a function determined by its expansion in a series in orthogonal polynomials. This criterion can be regarded as an analog of the well-known Kronecker criterion of rationality for functions given by power series.

### On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1145–1148

We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class $A_1$.

### Approximation of some classes of periodic functions of many variables

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1149–1152

We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations.