2019
Том 71
№ 11

# Mel'nik Yu. I.

Articles: 16
Brief Communications (Russian)

### On conditions for convergence of Taylor-Dirichlet series in a convex domain

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1576-1581

We establish necessary and sufficient conditions for the absolute convergence of the series $$\mathop \sum \limits_{v = 1}^\infty \sum\limits_{k = 0}^{m_v - I} {a_{v,k} z^k \exp (\lambda _v z)}$$ in an open region. We also give conditions under which an arbitrary function analytic in a closed region (analytic in an open region and continuous in a closed region) can be represented by a series of this type.

Article (Ukrainian)

### On the rate of convergence of double series of exponents representing regular functions on products of convex polygons

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1271–1274

Estimates exact in order are obtained in the uniform and integral metrics for the deviation of partial sums of double series of exponents that represent functions which are regular on products of convex polygons and either continuous on a product of closed polygons or belonging to the Smirnov class.

Brief Communications (Russian)

### On divergence of series of exponents representing functions regular in convex polygons

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 443–445

We prove that, on a convex polygon, there exist functions from the Smirnov class E whose series of exponents diverge in the metric of the space E. Similar facts are established for the convergence almost everywhere on the boundary of a polygon, for the uniform convergence on a closed polygon, and for the pointwise convergence at noncorner points of the boundary.

Brief Communications (Russian)

### On Green's function for the Helmholtz equation in a wedge

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1312-1314

It is found that, in the spherical coordinate system, the fundamental solution of the Helmholtz equation in a wedge satisfies the Sommerfeld radiation conditions at infinity uniformly in angle coordinates.

Article (Ukrainian)

### Inverse theorems for approximation of functions regular in convex polygons by exponential polynomials in the integral metric

Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 751-757

Article (Ukrainian)

### Direct theorems for the approximation of functions, regular in convex polygons, by exponential polynomials in the integral metric

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 584-591

Article (Ukrainian)

### Approximation of functions regular in convex polygons by exponential polynomials

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 446–452

Article (Ukrainian)

### Rate of convergence of exponential series representing functions regular in convex polygons

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 719–722

Article (Ukrainian)

### Absolute convergence of series of exponents that represent regular functions in convex polygons

Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 772–782

Article (Ukrainian)

### Uniqueness of expansion of a function, regular in a convex polygon, into a series of exponential functions

Ukr. Mat. Zh. - 1982. - 34, № 2. - pp. 217-219

Article (Ukrainian)

### A method of integral equations and the Riemann boundary problem

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 382–385

Article (Ukrainian)

### Approximate solution of integral equations in potential theory

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 385–391

Article (Ukrainian)

### Dirichlet series of functions, regular in convex polygons

Ukr. Mat. Zh. - 1980. - 32, № 6. - pp. 837–843

Article (Ukrainian)

### A universal system of exponentials

Ukr. Mat. Zh. - 1979. - 31, № 2. - pp. 192–196

Article (Ukrainian)

### Representation of regular functions by series of Mittag-Leffler functions in a closed disk

Ukr. Mat. Zh. - 1978. - 30, № 1. - pp. 114–120

Article (Ukrainian)

### Representation of regular functions by Dirichlet series in closed convex polygons

Ukr. Mat. Zh. - 1977. - 29, № 6. - pp. 826–830