2019
Том 71
№ 11

# Volume 15, № 4, 1963

Article (Russian)

### Existence of weak solutions of certain boundary value problems for equations of mixed type

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 347-364

The differential equation of mixed type $$Lu =\sum^2_{j, k=1}D_j (b_{jk} (x) D_ku) + \sum^2_{j=1}p_i(x)D_ju + p(x)u = f(x)$$ is considered in a bounded domain of the $(x_1, x_2)$-plane, the equation being for $x_2 > 0$ elliptic and for $x_2 < 0$ of the form $k(x_2) D^2_1u + D^2_2u = f(x)$. For boundary conditions of the Tricomi type, as well as for more general conditions, two energetic inequalities are proved (for the original and adjoint problems). The existence of the weak and the uniqueness of the strong solutions follows directly for the problems under consideration. Similar problems are investigated for certain unbounded domains.

Article (Russian)

### Converse theorems of the theory of approximation of functions in complex regions

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 365-375

An inequality is established for the modulus of the derivative of the algebraic polynomial $P_n(z)$ of degree $n$ to the effect that if, on an analytic arc $C$ on a piecewise-smooth boundary $C$ of a simply connected region $G$, $P_n(z)$ satisfies the condition $$|P_n(z)| \leq [\varrho_{l+1/n} (z)]^s \omega|\varrho_{l+1/n}(z)|, \quad(1)$$ where $\omega(t)$ is some modulus of continuity, $\varrho_{l+1/n}(z)$ is the distance from $z \in C$ to the $n$th line of level $C_n$ (i.e. to the line $\Phi(z) = R\left(1 + \cfrac1n\right)$, where $\Phi(z)$ is the mapping function of the outside $C$ on the outside of a unit circle, and $R$ is the conforming radius, $G$ and $s \leq 0$ then $$|P^1_n(z)| \leq A[\varrho_{l+1/n}(z)]^{s-1}\omega [\varrho_{l+1/n}(z)], A = const \quad(2)$$ After thjs an estimate is given of the continuity modulus of the rth derivative ($z$ is a whole number $\leq 0$) of the function $f(z)$ on $C$ under the condition that with each natural $n$ a polynomial $P,(z)$ can be found for it, such that $$|f(z) — P^1-n(z)| \leq [\varrho_{l+1/n}(z)]^r\omega [\varrho_{l+1/n}(z)]\quad(3)$$

Article (Russian)

### Unbounded operators commuting with an resolution of identity

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 376-385

In this paper the author gives a canonical form of a closed operator commuting with an resolution of identity (of infinite multiplicity) on a Hilbert space and proves the existence of a full system of generalized eigenfunctions for a related spectral operator.

Article (Russian)

### Spectral structure and self - conjugation of perturbations of differential operators with constant coefficients

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 385-399

If $P(D) + \sum^k_{j=1}c_j(x)Q_j(D)$ is a formally self-conjugate differential expression with sufficiently smooth and rapidly decreasing toward infinity coefficients $c_j(x)$ there exist absolutely summable throughout the space of derivatives $c_j(x)$ to the order $\max \left\{n + 1, q_j\right\}$, where $n$ is the dimension of the space, and $q_j$ is the degree of the polynomial $Q_j(\xi)$, and $$\lim_{[\eta]\rightarrow\infty}\int_{|\xi - \eta|\leq1}\frac{\sum^k_{j=1}|Q_j(\xi)|^2}{1 + |P(\xi)|^2}$$ the closure in $L_2(E^n)$ of the differential operator, determined on a class of infinitely differentiable finite functions $c^{\infty}_0$ by means of the differential exit pression $P(D) + \sum^k_{j=1}c_j(x) Q_j(D)$ is a self-conjugate operator, the limiting spectrum of which coincides with the set of values of the polynomial $P(\xi)$.

Article (Russian)

### Sur la question de construction effective des approximations tchebycheviennes du type rationnel fractionaire et de quelques types appraentés

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 400-411

Pour la méthode des interpolations tchebycheviennes successives ([8J; [9], § 5 dans le cas polynomial), qui s'étend pratiquement d'une manière assez satisfaisante au problème bien plus compliqué des représentations approximatives de la forme (2)—(3) aussi [1], [3], une variante est en passant signalée ici, libre de la procédure un peu pénible de résolution des systèmes d'équations linéaires. Cette méthode est confrontée avec une autre, de nature tout différente, que Ton peut nommer méthode des épreuves successives, récemment proposée fil], [12] pour les problèmes discrets des approximations tchebycheviennes de la forme (4'). On attire l'attention (§ 1°) au fait que l'idée de celle-ci reste aussi applicable aux représentations (8) — (5') analogues a (2) — (3) et l'on détermine explicitement (§ 2°) quelques procédures calculatoires, bonnes a la réalisation de cette méthode. Dans le dernier paragraphe 3° on compare les méthodes sur un exemple illustratif simple et l'on tire au clair certains aspects d'utilisation combinée possible des deux méthodes si différentes par leur nature.

Brief Communications (Russian)

### Linear methods of summing Fourier series and the best approximation

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 412-418

Brief Communications (Russian)

### On a class of functionals integrated by non-positive distributions

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 418-420

Brief Communications (Russian)

### A converse bondary value problem of the theory of filtration

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 420-427

Brief Communications (Russian)

### On the solution of the problems of free filtration by the method of successive approximations

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 427-431

Brief Communications (Russian)

### Basic boundary value problems of the theory of potential for regions with «slots»

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 431-437

Brief Communications (Russian)

### Integral representations of asign - variable group of the fourth degree

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 437-444

Brief Communications (Russian)

### Local increase in smoothness up to the boundary of solutions of elliptical equations

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 444-448

Brief Communications (Russian)

### On the categary of representations

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 448-453

Brief Communications (Russian)

### On the theory of finitely approximated groups

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 453-457

Letter (Russian)

### Letter to the editor

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 457-457

Index (Russian)

### Alphabetical in lex of the Ukrainian Mathematical Jornal, vol. XV, 1963

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 458-459