# Babenko V. F.

### One inequality of the Landau – Kolmogorov type for periodic functions of two variable

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 158-167

We obtain a new sharp inequality of the Landau – Kolmogorov type for a periodic function of two variables that estimates the convolution of the best uniform approximations of its partial primitives by the sums of univariate functions with the help of its $L_{\infty}$ -norm and uniform norms of its mixed primitives. Some applications of the obtained inequality are presented.

### Jackson – Stechkin-type inequalities for the approximation of elements of Hilbert spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1155-1165

We introduce new characteristics for elements of Hilbert spaces, namely, generalized moduli of continuity \$\omega_{ \varphi} (x, L_p, V ([0, \delta]))$ and obtain new exact Jackson – Stechkin-type inequalities with these moduli of continuity for the approximation of elements of Hilbert spaces. These results include numerous well-known inequalities for the approximation of periodic functions by trigonometric polynomials, approximation of nonperiodic functions by entire functions of exponential type, similar results for almost periodic functions, etc. Some of these results are new even in these classical cases.

### Estimation of the uniform norm of one-dimensional Riesz potential of a partial derivative of a function with bounded Laplacian

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2016. - 68, № 7. - pp. 867-878

We obtain new exact Landau-type estimates for the uniform norms of one-dimension Riesz potentials of the partial derivatives of a multivariable function in terms of the norm of the function itself and the norm of its Laplacian.

### On the optimal reconstruction of the convolution of $n$ functions according to the linear information

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 579-585

We determine the optimal linear information and the optimal method of its application to the recovery of convolution of $n$ functions on some convex and centrally symmetric classes of $2\pi$ -periodic functions.

### Approximation of some classes of set-valued periodic functions by generalized trigonometric polynomials

Babenko V. F., Babenko V. V., Polishchuk M. V.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 449-459

We generalize some known results on the best, best linear, and best one-sided approximations by trigonometric polynomials from the classes of $2 \pi$ -periodic functions presented in the form of convolutions to the case of classes of set-valued functions.

### On the Optimal Recovery of Integrals of Set-Valued Functions

Babenko V. F., Babenko V. V., Polishchuk M. V.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1163-1171

We consider the problem of optimization of the approximate integration of set-valued functions from the class specified by a given majorant of their moduli of continuity performed by using the values of these functions at *n* fixed or free points of their domain.

### Motornyi Vitalii Pavlovych (on his 75th birthday)

Babenko V. F., Davydov O. V., Kofanov V. A., Parfinovych N. V., Pas'ko A. N., Romanyuk A. S., Ruban V. I., Samoilenko A. M., Shevchuk I. A., Shumeiko A. A., Timan M. P., Trigub R. M., Vakarchuk S. B., Velikin V. L.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999

### Optimal Recovery of *n*-Linear Functionals According to Linear Information

Babenko V. F., Gun’ko M. S., Rudenko A. A.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 884–890

We determine the optimal linear information and the optimal procedure of its application for the recovery of *n*-linear functionals on the sets of special form from a Hilbert space.

### Major Pylypovych Timan (on his 90th birthday)

Babenko V. F., Motornyi V. P., Peleshenko B. I., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Trigub R. M., Vakarchuk S. B.

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1141-1144

### Uniformly distributed ridge approximation of some classes of harmonic functions

Babenko V. F., Levchenko D. A.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1426-1431

We determine the exact values of the uniformly distributed ridge approximation of some classes of harmonic functions of two variables.

### Approximation of some classes of functions of many variables by harmonic splines

Babenko V. F., Leskevich T. Yu.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1011-1024

We determine the exact values of upper bounds of the error of approximation by harmonic splines for functions $u$ defined on an $n$-dimensional parallelepiped $\Omega$ forwhich $||\Delta u||_{L_{\infty}(\Omega)} \leq 1$ and for functions $u$ defined on $\Omega$ forwhich $||\Delta u||_{L_{p}(\Omega)} \leq 1, \quad 1 \leq p \leq \infty$. In the first case, the error is estimated in $L_{p}(\Omega), \quad 1 \leq p \leq \infty$; in the second case, it is estimated in $L_{1}(\Omega)$.

### On the dependence of the norm of a function on the norms of its derivatives of orders $k$ , $r - 2$ and $r , 0 < k < r - 2$

Babenko V. F., Kovalenko O. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 597-603

We establish conditions for a system of positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}, \; 0 = k_1 < k2 < k3 = r − 2, k4 = r$, necessary and sufficient for the existence of a function $x \in L^r_{\infty, \infty}(R)$ such that $||x^{(k_i)} ||_{\infty} = M_{k_i},\quad i = 1, 2, 3, 4$.

### Estimates for the norms of fractional derivatives in terms of integral moduli of continuity and their applications

Babenko V. F., Churilova M. S.

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1155-1168

For functions defined on the real line or a half-line, we obtain Kolmogorov-type inequalities that estimate the $L_p$-norms $(1 \leq p < \infty)$ of fractional derivatives in terms of the Lp-norms of functions (or the $L_p$-norms of their truncated derivatives) and their $L_p$-moduli of continuity and establish their sharpness for $p = 1$. Applications of the obtained inequalities are given.

### Bernstein-type inequalities for splines defined on the real axis

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 603-611

We obtain the exact inequalities of the Bernstein type for splines $s \in S_{m, h} \bigcap L_2 (\mathbb{R})$ as well as the exact inequalities that, for splines $s \in S_{m, h}, \quad h > 0$, estimate $L_p$-norms of the Fourier transforms of their $k$-th derivative by $L_p$-norms of the Fourier transforms of splines themselves.

### Optimization of approximate integration of set-valued functions monotone with respect to inclusion

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 147-155

The best quadrature formula is found for the class of convex-valued functions defined on the interval [0, 1] and monotone with respect to an inclusion.

### Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Babenko V. F., Parfinovych N. V., Pichugov S. A.

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 301–314

Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented.

### On the order of relative approximation of classes of differentiable periodic functions by splines

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 147–157

In the case where $n → ∞$, we obtain order equalities for the best $L_q$ -approximations of the classes $W_p^r ,\; 1 ≤ q ≤ p ≤ 2$, of differentiable periodical functions by splines from these classes.

### Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1443-1454

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.

### Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space

Babenko V. F., Bilichenko R. O.

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1299-1305

The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the $L_2$-norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.

### Inequalities of the Bernstein type for splines of defect 2

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 995-999

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order *r* and defect 2.

### Approximation of unbounded operators by bounded operators in a Hilbert space

Babenko V. F., Bilichenko R. O.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 147-153

We determine the best approximation of an arbitrary power $A^k$ of an unbounded self-adjoint operator $A$ in a Hilbert space $H$ on the class $\{x ∈ D(A^r ) : ∥A^r x∥ ≤ 1\},\; k < r$.

### On the best *L*_{2 }-approximations of functions by using wavelets

Babenko V. F., Zhiganova G. S.

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1119 – 1127

We obtain the exact Jackson-type inequalities for approximations in *L*_{2 }(*R*) of functions *f*∈ *L*_{2 }(*R*)
with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon–Kotelnikov wavelets.

### Problem of uniqueness of an element of the best nonsymmetric *L*_{ 1}-approximation of continuous functions with values in *KB* -spaces

Babenko V. F., Tkachenko M. E.

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 867 – 878

### On Kolmogorov-type inequalities for fractional derivatives of functions of two variables

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 837–842

We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order.

### Estimates for wavelet coefficients on some classes of functions

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1594–1600

Let $ψ_m^D$ be orthogonal Daubechies wavelets that have $m$ zero moments and let $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. We prove that $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$

### On Kolmogorov-type inequalities for functions defined on a semiaxis

Babenko V. F., Skorokhodov D. S.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1299–1312

Necessary and sufficient conditions for the existence of a function from the class *S*^{ -}
with prescribed values of integral norms of three successive derivatives (generally speaking, of a fractional order) are obtained.

### Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

Babenko V. F., Churilova M. S.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 597–606

We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, where $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$ is the exact constant in a Kolmogorov-type inequality, $ℝ$ is the real straight line, $T = [0,2π],\; L^l_{p, p} (G^n)$ is the set of functions $ƒ ∈ L_p (G^n)$ such that the partial derivative $D_i^{l_i } f(x)$ belongs to $L_p (G^n), i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$ is the mixed derivative of a function $ƒ, 0 < µi < 1, i = \overline {0,n},$ and $∑_{i=0}^n µ_i = 1$. If $G^n = ℝ$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$ if $G^n = T^n$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i= \overline {1,n},\; λ ≥ 0$. We prove that, for $λ = 0$, the equality $K(ℝ^n) = K(T^n)$ is true.

### Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 291–302

We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$ and for functions of constant sign on $R$ or $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$ and in the case where $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$

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

Babenko V. F., Doronin V. G., Ligun A. A., Shumeiko A. A.

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 $L_2$.

### Approximation of sine-shaped functions by constants in the spaces $L_p,\; p < 1$

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 745–762

We investigate the best approximations of sine-shaped functions by constants in the spaces $L_p$ for $p < 1$. In particular, we find the best approximation of perfect Euler splines by constants in the spaces Lp for certain $p∈(0,1)$.

### Kolmogorov-type inequalities for mixed derivatives of functions of many variables

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

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 579-594

Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ for all $x \in L^{r\gamma}_{\infty}(T^d)$ Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$

### Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 579-589

We investigate the relationship between the constants *K*(**R**) and *K*(**T**), where \(K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}\) is the exact constant in the Kolmogorov inequality, **R** is the real axis, **T** is a unit circle, $$L_{p,s}^r (G)$$ is the set of functions *x* ∈ *L* _{p}(*G*) such that *x* ^{(r)} ∈ *L* _{s}(*G*), *q*, *p*, *s* ∈ [1, ∞], *k*, *r* ∈ **N**, *k* < *r*, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$
then*K*(*R*) = *K*(*T*),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$
then*K*(**R**) ≤ *K*(**T**); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis.

### On Kolmogorov-Type Inequalities with Integrable Highest Derivative

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1694-1697

We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions \(x \in L_1^r\) and any *k*, *r* ∈ *N*, *k* < *r*. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of *K*-functional type.

### Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 603-609

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions *x* with bounded variation of the derivative *x*′, namely $$\left\| {x'} \right\|_q \leqslant K\left( {q,p} \right)\left\| x \right\|_p^a \left( {\mathop V\limits_{0}^{{2\pi }} \left( {x'} \right)} \right)^{1 - {alpha }} ,$$ where *q* ∈ (0, ∞), *p* ∈ [1, ∞], and α = min{1/2, *p*/*q*(*p* + 1)}.

### Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1299-1308

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for *r* = 2, *k* = 1 or *r* = 3, *k* = 1, 2 and arbitrary *q, p* ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ_{ r } is the perfect Euler spline of order *r*.

### 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.

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

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

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

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

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

### 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.

### 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.

### 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.

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.

### Optimization of approximate integration of monotone functions of two variables

Babenko V. F., Borodachov S.V.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 881–889

We solve the problem of optimization of approximate integration of functions of two variables defined on a rectangle and monotonic in each variable by using the quadrature formulas with nodes at points of a rectangle net.

### On the best $L_1$-approximations of functional classes by splines under restrictions imposed on their derivatives

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 435-444

We find the exact asymptotics ($n → ∞$) of the best $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s ∈ S_{2n, r∼-1}$ ($S_{2n, r∼-1}$ is a set of $2π$-periodic polynomial splines of order $r−1$, defect one, and with nodes at the points $kπ/n,\; k ∈ ℤ$) such that $V_0^{2π} s^{( r-1)} ≤ 1+ɛ_n$, where $\{ɛ_n\}_{n=1}^{ ∞}$ is a decreasing sequence of positive numbers such that $ɛ_n n^2 → ∞$ and $ɛ_n → 0$ as $n → ∞$.

### On the connection between certain inequalities of the Kolmogorov type for periodic and nonperiodic functions

Babenko V. F., Selivanova S. A.

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 147–157

We obtain nonperiodic analogs of the known inequalities that estimate*L* _{ p }-norms of intermediate derivatives of a periodic function in terms of its*L* _{∞}-norms and higher derivative.

### On exact constants in inequalities for norms of derivatives on a finite segment

Babenko V. F., Uedraogo Zh. B.

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 117–119

We prove that, in an additive inequality for norms of intermediate derivatives of functions defined on a finite segment and equal to zero at a given system of points, the least possible value of a constant coefficient of the norm of a function coincides with the exact constant in the corresponding Markov-Nikol'skii inequality for algebraic polynomials that are also equal to zero at this system of points.

### The best $L_1$-approximations of classes of functions defined by differential operators in terms of generalized splines from these classes

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1443-1451

For classes of periodic functions defined by constraints imposed on the $L_1$-norm of the result of action of differential operators with constant coefficients and real spectrum on these functions, we determine the exact values of the best $L_1$-approximations by generalized splines from the classes considered.

### Comparison of approximation properties of generalized polynomials and splines

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1011–1020

We establish that, for *p* ∈ [2, ∞), *q* = 1 or *p* = ∞, *q ∈* [ 1, 2], the classes *W* _{p} ^{r} of functions of many variables defined by restrictions on the *L* _{p}-norms of mixed derivatives of order *r* = (*r* _{1}, *r* _{2}, ..., *r* _{m}) are better approximated in the *L* _{q}-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide.

### On inequalities of Kolmogorov-Hörmander type for functions bounded on a discrete net

Babenko V. F., Vakarchuk M. B.

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 988–992

We obtain a strengthened version of the Hörmander inequality for functions ƒ: ℝ → ℝ, in which, instead of ‖ƒ‖_{∞}, we use the least upper bound of the values of *f* on a discrete set of points.

### On the optimal renewal of bilinear functionals in linear Normed spaces

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 828–831

We study the problem of optimal renewal of bilinear functionals on the basis of optimal linear information in the general statement. We also represent some new results for special spaces of functions.

### On additive inequalities for intermediate derivatives of functions given on a finite interval

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 619–628

We present a general scheme for deducing additive inequalities of Landau-Hadamard type. As a consequence, we prove several new inequalities for the norms of intermediate derivatives of functions given on a finite interval with an exact constant with the norm of a function.

### On the optimization of approximate integration by Monte Carlo methods

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 475–480

We solve the problem of optimization of monte Carlo methods for approximate integration over an arbitrary absolutely continuous measure. We propose a convenient model of Monte Carlo methods which uses the notion of transition probability.

### On optimization of weight quadrature formulas

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1011–1021

We obtain asymptotically optimal quadrature formulas on the class*H* ^{ ω }[-1, 1] for an arbitrary continuous weight function which is positive on [-1, 1] almost everywhere and for a wide class of moduli of continuity ω(*t*).

### Generalization of some extremal properties of splines

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 403–407

We generalize well-known inequalities for the norms of the derivatives of periodic splines with minimal defect, perfect splines, and monosplines.

### On inequalities for norms of intermediate derivatives on a finite interval

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 105–107

For functions*f* which have an absolute continuous (*n*−1)th derivative on the interval [0, 1], it is proved that, in the case of*n*>4, the inequality $$\left\| {f^{(n - 2)} } \right\|_\infty \leqslant 4^{n - 2} (n - 1) ! \left\| f \right\|_\infty + \left\| {f^{(n)} } \right\|_\infty /2$$ holds with the exact constant 4^{ n−2}(n−1)!.

### Approximation of continuous vector functions

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1435–1448

We study the possibility of uniform approximation of continuous mappings of metric compact sets into metric spaces. Notions of “weak dimension” and “weak Kolmogorov width” are introduced to compare approximating properties of infinite-dimensional subspaces. For classes of mappings specified by the majorant of the modulus of continuity, we present bilateral estimates of “weak” widths that may coincide under certain conditions.

### BestL1-approximations of classes $W_1^r$ by Splines from $W_1^r$

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1410–1413

We obtain the exact values of the best $L_1$-approximations of the classes $W_1^r$ of periodic functions by periodic polynomial splines of degree $r$ and defect 1 with equidistant knots that belong to the class $W_1^r$.

### On the uniqueness of elements of the best approximation and the best one-sided approximation in the space $L_1$

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 475–483

The problem of uniqueness of the best approximations in the space $L_1[a, b]$ is studied We consider the problem of the best approximation and the best $(\alpha, \beta)$-approximation of continuous functions and the problem of the best one-sided approximation of continuously differentiable functions.

### Some Bernstein-type inequalities for trigonometric polynomials

Ukr. Mat. Zh. - 1993. - 45, № 3. - pp. 428–430

The refinements of some well-known Bemstein-type inequalities for trigonometric polynomials are obtained.

### Bernstein-type inequalities for $\mathcal{L}$-splines

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 10-20

New Berhstein-type inequalities are obtain for 2π-periodic $\mathcal{L}$ -splines associated with a differential operator $\mathcal{L}_\tau (D)$ of degree $r$ with fixed real coefficients.

### Optimal reconstruction of convolutions and scalar products of functions from various classes

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1305–1310

### Widths and optimal quadrature formulas for convolution classes

Ukr. Mat. Zh. - 1991. - 43, № 9. - pp. 1135–1148

### Development of studies on the exact solution of extremal problems of the theory of best approximation

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 4–17

### Approximate computation of scalar products

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 15-21

### Sharp inequalities for the norms of conjugate functions and their applications

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 139-144

### Inequalities for the derivatives of polynomials with real zeros

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 411–416

### Widths of certain classes of convolutions

Ukr. Mat. Zh. - 1983. - 35, № 5. - pp. 603—607

### Nonsymmetric approximation in spaces of integrable functions

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 409—416

### A property of compact operators in the space of integrable functions

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 491–492

### Approximation in the mean of linear combinations of shifts of certain functions

Ukr. Mat. Zh. - 1981. - 33, № 2. - pp. 234–240