2019
Том 71
№ 2

All Issues

Volume 52, № 1, 2000

Article (Russian)

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

Babenko V. F., Ligun A. A., Mitropolskiy Yu. A., Motornyi V. P., Nikol'skii S. M., Samoilenko A. M.

Full text (.pdf)

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

Article (Russian)

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

Babenko V. F., Ligun A. A., Motornyi V. P.

↓ Abstract   |   Full text (.pdf)

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

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

Article (Russian)

Investigations of dnepropetrovsk mathematicians related to inequalities for derivatives of periodic functions and their applications

Babenko V. F.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 9-29

We present a survey of investigations of Dnepropetrovsk mathematicians related to Kolmogorov-type exact inequalities for norms of intermediate derivatives of periodic functions and their applications in approximation theory.

Article (Russian)

On the uniqueness of an element of the best $L_1$-approximation for functions with values in a banach space

Babenko V. F., Gorbenko M. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 30-34

We study the problem of uniqueness of an element of the best $L_1$-approximation for continuous functions with values in a Banach space. We prove two theorems that characterize the uniqueness subspaces in terms of certain sets of test functions.

Article (Russian)

On the best approximation in the mean and overconvergence of a sequence of polynomials of the best approximation

Vakarchuk S. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 35-45

We investigate one property of a sequence of polynomials of the best approximation in the mean related to the convergence in a neighborhood of every point of regularity of a function on the level line ∂ G R.

Article (Russian)

Exact constants in inequalities of the jackson type for quadrature formulas

Doronin V. G., Ligun A. A.

↓ Abstract   |   Full text (.pdf)

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 the best approximation of periodic functions of two variables by polynomial splines

Korneichuk N. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 52-57

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.

Article (Russian)

Inequalities for polynomial splines

Korneichuk N. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 58-65

We establish exact estimates for the variation on a period of the derivative s (r)(t) of a periodic polynomial spline s(t) of degree r and defect 1 with respect to a fixed partition of [0, 2π) under the condition that \(\left\| {s^{(r)} } \right\|_X = 1\) , where X=C or L 1

Article (Russian)

Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications

Babenko V. F., Kofanov V. A., Korneichuk N. P., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 66-84

We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes $W^r H^{ω}$, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class $H^{ω}$ on the derivatives of trigonometric polynomials or polynomial splines in terms of the $L^{ϱ}$ -norms of these polynomials and splines.

Article (Russian)

On asymptotically exact estimates for the approximation of certain classes of functions by algebraic polynomials

Motornaya O. V., Motornyi V. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 85-99

We present a survey of results obtained for the last decade in the field of approximation of specific functions and classes of functions by algebraic polynomials in the spaces C and L 1 and approximation with regard for the location of a point on an interval.

Article (Ukrainian)

Isogeometric spline reconstruction of plane curves

Nazarenko N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 100-105

We establish conditions for the isogeometric reconstruction of plane curves by using parabolic and cubic parametric splines of minimal defect.

Article (Russian)

Optimal discretization of Ill-posed problems

Pereverzev S. V., Solodkii S. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 106-121

We present a survey of results on the optimal discretization of ill-posed problems obtained in the Institute of Mathematics of the Ukrainian National Academy of Sciences.

Article (Russian)

On the jackson theorem for periodic functions in spaces with integral metric

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 122-133

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L p , 0 < p < 1, and L 0. In particular, we prove the multidimensional Jackson theorem in L p (T m ), 0 < p < 1.

Article (Russian)

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

Shumeiko A. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 134-144

For functions integrable to the power \(\beta = (r + 1 + 1/p)^{ - 1} \) , we obtain asymptotically exact lower bounds for the approximation by local splines of degree r and defect k< r/2 in the metric of L p