2018
Том 70
№ 9

# Ligun A. A.

Articles: 22
Article (Russian)

### Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 92-98

We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.

Article (Russian)

### Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 92-98

We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.

Article (Russian)

### On Jackson-Type Inequalities for Functions Defined on a Sphere

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 291–304

We obtain exact estimates of the approximation in the metrics $C$ and $L_2$ of functions, that are defined on a sphere, by means of linear methods of summation of the Fourier series in spherical harmonics in the case where differential and difference properties of functions are defined in the space $L_2$.

Article (Russian)

### Recovery of a Function from Information on Its Values at the Nodes of a Triangular Grid Based on Data Completion

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 332-341

We consider a method for binary completion of two-dimensional data. On the basis of information about a surface given by a triangular grid, we construct a continuous polygonal surface based on a denser grid (than the one given). We determine the error and norm of this method and study its properties.

Article (Ukrainian)

### A Linear Method for the Recovery of Functions Based on Binary Data Completion

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1501-1512

We construct a linear recovery method based on binary data completion using the Bessel interpolation formula. We find an asymptotic value of the error of this method, determine its norm, and study its properties.

Article (Russian)

### Description of Convex Curves

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 908-922

We present a description of convex curves, which enables one to reduce the problem of approximation of a convex curve by piecewise circular lines in the Hausdorff metric to the problem of approximation of 2π-periodic functions by trigonometric splines in the uniform metric. We describe certain properties of convex curves.

Article (Russian)

### 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.

Article (Russian)

### On the 80th birthday of Academician N. P. Korneichuk

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 3-4

Article (Russian)

### On the results of N. P. Korneichuk obtained in 1990–1999

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 5-8

We present a brief survey of Korneichuk’s works published in 1990–1999.

Article (Russian)

### Exact constants in inequalities of the jackson type for quadrature formulas

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 46-51

We prove that if $R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)$ is the error of a simple quadrature formula and ω(ε, δ)1 is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: $\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{\omega (f^{(r)} ,\delta )_1 }} = \frac{{\pi \left\| {D_1 } \right\|_\infty }}{{n^r }}$ whereD r is the Bernoulli kernel.

Article (Russian)

### On lower bounds for the approximation of individual functions by local splines with nonfixed nodes

Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1628–1637

For functions with the integrable βth power, where β = (r + 1 + 1/p)−1, we obtain asymptotically exact lower bounds for the approximation by local splines of degreer and defectkr/2 in the metric ofL p.

Article (Russian)

### On a problem of restoration of curves

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 181–189

We investigate the problem of asymptotically optimal placement of disks of the same radius under the condition of minimization of the Hausdorff distance between a given curve $Γ$ and the union of disks under study.

Article (Russian)

### On extremal problems on classes of functions defined by integral moduli of continuity

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1499–1503

We obtain lower bounds for solutions of some extremal problems on classes of functions W rH 1 ω with integral modulus of continuity ω(t). Some of these bounds are regarded as exact.

Article (Russian)

### On one problem of minimization of area

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 927–936

We consider the problem of asymptotically optimal location of disks with equal radii for the minimization of the are of the figure bounded by a given curve and a connected union of these disks.

Article (Ukrainian)

### Exact constant in the Jackson inequality in the space L2

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1261–1265

Article (Ukrainian)

### Duality for L-splines

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 776-778

Article (Ukrainian)

### Asymptotically best quadrature formulas on classes of differentiable functions

Ukr. Mat. Zh. - 1983. - 35, № 1. - pp. 94—98

Article (Ukrainian)

### Approximation of differentiable periodic functions by local splines of minimum deficiency

Ukr. Mat. Zh. - 1981. - 33, № 5. - pp. 691—693

Article (Ukrainian)

### Error bound of spline interpolation in an integral metric

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 391–394

Article (Ukrainian)

### Best choice of knots in approximation of functions by local hermitian splines

Ukr. Mat. Zh. - 1980. - 32, № 6. - pp. 824–830

Article (Ukrainian)

### A property of interpolational spline functions

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 507–514

Article (Ukrainian)

### Approximation of periodic functions by splines of minimal defect

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 388 – 392