2017
Том 69
№ 12

# Babenko V. F.

Articles: 49
Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

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

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.

Article (English)

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

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.

Anniversaries (Ukrainian)

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

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

Article (Russian)

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

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.

Anniversaries (Ukrainian)

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

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

Brief Communications (Russian)

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

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.

Article (Russian)

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

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 $L_{1}(\Omega)$.

Article (Russian)

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

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 $||x^{(k_i)} ||_{\infty} = M_{k_i},\quad i = 1, 2, 3, 4$.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

### 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.

Article (English)

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

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.

Article (Russian)

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

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.

Article (Russian)

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

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.

Article (Russian)

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

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.

Brief Communications (Russian)

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

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.

Article (Russian)

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

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 $\{x ∈ D(A^r ) : ∥A^r x∥ ≤ 1\},\; k < r$.

Brief Communications (Russian)

### On the best L2 -approximations of functions by using wavelets

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

We obtain the exact Jackson-type inequalities for approximations in L2 (R) of functions fL2 (R) with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon–Kotelnikov wavelets.

Article (Ukrainian)

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

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

Brief Communications (Russian)

### 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.

Article (Russian)

### 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 $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$

Article (Russian)

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

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.

Article (Russian)

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

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.

Article (Russian)

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

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 $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$

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)

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

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 $p∈(0,1)$.

Article (Russian)

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

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 $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$

Article (Russian)

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

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 xL p(G) such that x (r)L s(G), q, p, s ∈ [1, ∞], k, rN, k < r, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$ thenK(R) = K(T),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$ thenK(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.

Brief Communications (Russian)

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

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, rN, 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.

Article (Russian)

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

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)}.

Article (Ukrainian)

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

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.

Brief Communications (Russian)

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

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.

Brief Communications (Russian)

### 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.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Ukrainian)

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

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

Article (Ukrainian)

### On the uniqueness of elements of the best approximation and the best one-sided approximation in the space L1

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

Article (Ukrainian)

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

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

Article (Ukrainian)

### Widths and optimal quadrature formulas for convolution classes

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

Article (Ukrainian)

### 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

Article (Ukrainian)

### Approximate computation of scalar products

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

Article (Ukrainian)

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

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

Article (Ukrainian)

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

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

Article (Ukrainian)

### Widths of certain classes of convolutions

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

Article (Ukrainian)

### Nonsymmetric approximation in spaces of integrable functions

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

Article (Ukrainian)

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

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

Article (Ukrainian)

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

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