2019
Том 71
№ 2

# Volume 52, № 1, 2000

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)

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

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

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

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

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

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

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

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

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

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

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

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

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