# Vakarchuk S. B.

### On the estimates of widths of the classes of functions defined by the generalized moduli of continuity and majorants in the weighted space $L_{2x} (0,1)$

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 179-189

The upper and lower estimates for the Kolmogorov, linear, Bernstein, Gelfand, projective, and Fourier widths are obtained in the space $L_{2,x}(0, 1)$ for the classes of functions $W^r_2 (\Omega^{(\nu )}_{m,x}; \Psi )$, where $r \in Z+, m \in N, \nu \geq 0,$ and $\\Omega^{(\nu )}_{m,x}$ and $\Psi$ are the mth order generalized modulus of continuity and the majorant, respectively. The upper and lower estimates for the suprema of Fourier – Bessel coefficients were also found on these classes. We also present the conditions for majorants under which it is possible to find the exact values of indicated widths and the suprema of Fourier – Bessel coefficients.

### Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space $L_2 (R)$. II

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1345-1373

In the second part of the paper, we establish the exact Jackson-type inequalities for the characteristic of smoothness $\Lambda^w$ on the classes of functions $L^{\alpha}_2 (R)$ defined by the fractional derivatives of order $\alpha \in (0,\infty )$ in the space $L_2(R)$. The exact values of the mean $\nu$ -widths for the classes of functions, defined by the generalized characteristics of smoothness $\omega w$ and $\Lambda w$ are also computed in $L_2(R)$.

### Generalized characteristics of smoothness and some extremе problems of the approximation theory of functions in the space $L_2 (R)$. I

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1166-1191

We consider the generalized characteristics of smoothness of the functions $\omega^w(f, t)$ and $\Lambda^w(f, t), t > 0,$ in the space $L_2(R)$ and, on the classes $L^{\alpha}_2 (R)$ defined with the help of fractional-order derivatives $\alpha \in (0,\infty)$, obtain the exact Jackson-type inequalities for $\omega^w(f)$.

### On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximations by entire functions of the exponential type on the entire real axis

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 599-623

The exact Jackson-type inequalities with modules of continuity of a fractional order $\alpha \in (0,\infty )$ are obtained on the classes of functions defined via the derivatives of a fractional order $\alpha \in (0,\infty )$ for the best approximation by entire functions of the exponential type in the space $L_2(R)$. In particular, we prove the inequality $$2^{- \beta /2}\sigma^{- \alpha} (1 - \cos t)^{- \beta /2} \leq \sup \{ \scr {A}_\sigma (f) / \omega_{\beta }(\scr{D}^{\alpha} f, t/\sigma ) : f \in L^{\alpha}_2 (R)\} \leq \sigma^{-\alpha} (1/t^2 + 1/2)^{\beta /2},$$ where $\beta \in [1,\infty ), t \in (0, \pi ], \sigma \in (0,\infty ).$ The exact values of various mean $\nu$ -widths of the classes of functions determined via the fractional modules of continuity and majorant satisfying certain conditions are also determined.

### On total moduli of continuity for $2\pi$-periodic functions of two variables in the space $L_{2,2}$

Vakarchuk M. B., Vakarchuk S. B.

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 300-310

The description of the total moduli of continuity of $2\pi$ -periodic functions of two variables are obtained in the space $L_{2,2}$. The proposed description can be regarded as an extension of the famous results by O. V. Besov, S. B. Stechkin, V. A.Yudin on the moduli of continuity in $L_{2}$ in the two-dimensional case.

### Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths of the classes of $(ψ,β)$-differentiable functions in $L_2$. II

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1021-1036

In the space $L_2$, we determine the exact values of some $n$-widths for the classes of functions such that the generalized moduli of continuity of their $(\psi, \beta)$ - derivatives or their averages with weight do not exceed the values of the majorants $\Phi$ satisfying certain conditions. Specific examples of realization of the obtained results are also analyzed.

### Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths of the classes of $(ψ,β)$-differential functions in $L_2$. I

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 723-745

For the generalized moduli of continuity, including the ordinary moduli of continuity and various their modifications, we establish the exact constants for Jackson-type inequalities in the classes of $2\pi$ -periodic functions in the space $L_2$ with $(\psi , \beta)$-derivatives, introduced by Stepanets. These results take into account the classification of $(\psi , \beta)$-derivatives and enable us to consider the major part of Jackson-type inequalities obtained earlier in the classes of differentiable functions $L_2^r,\; r \in N$, from the common point of view.

### 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

### On the Best Polynomial Approximations of Entire Transcendental Functions of Many Complex Variables in Some Banach Spaces

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1598–1614

For the entire transcendental functions $f$ of many complex variables $m (m ≥ 2)$ of finite generalized order of growth $ρ_m (f; α, β)$, we obtain the limiting relations between the indicated characteristic of growth and the sequences of best polynomial approximations of $f$ in the Hardy Banach spaces $H q (U^m )$ and in the Banach spaces $Bm(p, q, ⋋)$ studied by Gvaradze. The presented results are extensions of the corresponding assertions made by Varga, Batyrev, Shah, Reddy, Ibragimov, and Shikhaliev to the multidimensional case.

### Jackson-Type Inequalities for the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean $ν$ - Widths for the Classes of Functions in the Space $L_2 (ℝ)$

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 740–766

The exact values of constants are obtained in the space $L_2 (ℝ)$ for the Jackson-type inequalities for special moduli of continuity of the $k$ th order defined by the Steklov operator $S_h (f)$ instead of the translation operator $T_h (f)$ in the case of approximation by entire functions of exponential type $σ ∈ (0,∞)$. The exact values of the mean $ν$-widths (linear, Bernstein, and Kolmogorov) are also obtained for the classes of functions defined by the indicated characteristic of smoothness.

### On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions

Shvachko A. V., Vakarchuk S. B.

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1604–1621

The exact value of the extremal characteristic

is obtained on the class *L* _{2} ^{ r } (*D* _{ ρ })*,* where *r ∈* ℤ_{+}; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \) *, σ* and τ are polynomials of at most the second and first degrees, respectively, *ρ* is a weight function, 0 *< p* ≤ 2*,* 0 *< h <* 1*, λ* _{ n }(*ρ*)
are eigenvalues of the operator *D* _{ ρ } *, φ*
is a nonnegative measurable and summable function (in the interval (*a, b*)) which is not equivalent to zero, *Ω* _{ k,ρ } is the generalized modulus of continuity of the *k* th order in the space *L* _{2,ρ } (*a, b*)*,* and *E* _{ n } (*f*)_{2,ρ }
is the best polynomial approximation in the mean with weight *ρ* for a function *f ∈ L* _{2,ρ } (*a, b*)*.* The exact values of widths for the classes of functions specified by the characteristic of smoothness *Ω* _{ k,ρ } and the *K*-functional \( \mathbb{K} \) _{m} are also obtained.

### 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

### On the best polynomial approximation in the space L2 and widths of some classes of functions

Vakarchuk S. B., Zabutnaya V. I.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1025-1032

We consider the problem of the best polynomial approximation of $2\pi$-periodic functions in the space $L_2$ in the case where the error of approximation $E_{n-1}(f)$ is estimated in terms of the $k$th-order modulus of continuity $\Omega_k(f)$ in which the Steklov operator $S_h f$ is used instead of the operator of translation $T_h f (x) = f(x + h)$. For the classes of functions defined using the indicated smoothness characteristic, we determine the exact values of different $n$-widths.

### Best mean-square approximation of functions defined on the real axis by entire functions of exponential type

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 604-615

Exact constants in Jackson-type inequalities are calculated in the space $L_2 (\mathbb{R})$ in the case where the quantity of the best approximation $\mathcal{A}_{\sigma}(f)$ is estimated from above by the averaged smoothness characteristic $\Phi_2(f, t) = \cfrac 1t \int^t_0||\Delta^2_h(f)||dh$. We also calculate the exact values of the average $\nu$-widths of classes of functions defined by $\Phi_2$.

### On the exponential decay of vibrations of damped elastic media

Vakarchuk M. B., Vakarchuk S. B.

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1579-1601

Exact inequalities of the Kolmogorov type are obtained in Hardy Banach spaces for functions of one complex variable analytic in the unit disk and functions of two complex variables analytic in the unit bidisk. We also present applications of these inequalities to problems of the theory of approximation of analytic functions of one and two complex variables.

### Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes

Doronin V. G., Vakarchuk S. B.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1032–1043

We obtain exact Jackson-type inequalities in the case of the best mean square approximation by entire functions of finite degree $≤ σ$ on a straight line. For classes of functions defined via majorants of averaged smoothness characteristics $Ω_1(f, t ),\; t > 0$, we determine the exact values of the Kolmogorov mean ν-width, linear mean ν-width, and Bernstein mean $ν$-width, $ν > 0$.

### On the best polynomial approximation of entire transcendental functions of generalized order

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1011–1026

We prove a Hadamard-type theorem which connects the generalized order of growth $\rho^*_f(\alpha, \beta)$ of entire transcendental function $f$ with coefficients of its expansion into the Faber series. The theorem is an original extension of a certain result by S. K. Balashov to the case of finite simply connected domain $G$ with the boundary $\gamma$ belonging to the S. Ya. Al'per class $\Lambda^*.$
This enables us to obtain boundary equalities that connect $\rho^*_f(\alpha, \beta)$ with the sequence of the best polynomial approximations of $f$ in some Banach spaces of functions analytic in $G$.

### On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$

Shchitov A. N., Vakarchuk S. B.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 303-316

We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$ are calculated.

### Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines

Myskin K. Yu., Vakarchuk S. B.

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 147–157

Exact estimates for the errors of approximation of functions of two variables and their derivatives by interpolation bilinear splines are obtained on certain classes.

### Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions

Shchitov A. N., Vakarchuk S. B.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1458-1466

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions $f(x) ∈ L_2^r(r ∈ ℤ_{+})$ and expressions containing moduli of continuity of the $k$th order $ω_k(f^{(r)}, t)$. Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from $L_2$. For the classes $F (k, r, Ψ*)$ defined by $ω_k(f^{(r)}, t)$ and the majorant $Ψ(t)=t^{4k/π^2}$, we determine the exact values of different widths in the space $L_2$.

### On some extremal problems of approximation theory in the complex plane

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1155-1171

In the Hardy Banach spaces *H* _{ q }, Bergman Banach spaces *H*′_{q}, and Banach spaces ℬ (*p, q*, λ), we determine the exact values of the Kolmogorov, Bernstein, Gel’fand, linear, and trigonometric *n*-widths of classes of functions analytic in the disk |z| < 1 and such that the averaged moduli of continuity of their *r*-derivatives are majorized by a certain function. For these classes, we also consider the problems of optimal recovery and coding of functions.

### Jackson-type inequalities and exact values of widths of classes of functions in the spaces $S^p , 1 ≤ p < ∞$

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 595–605

In the spaces $S^p , 1 ≤ p < ∞$, introduced by Stepanets, we obtain exact Jackson-type inequalities and compute the exact values of widths of classes of functions determined by averaged moduli of continuity of order $m$.

### On the Best Polynomial Approximations of $2π$-Periodic Functions and Exact Values of $n$-Widths of Functional Classes in the Space $L_2$

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1603-1615

To solve extremal problems of approximation theory in the space $L_2$, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of $n$-widths of functional classes determined by these moduli.

### On Some Problems of Polynomial Approximation of Entire Transcendental Functions

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1155-1162

For entire transcendental functions of finite generalized order, we obtain limit relations between the growth characteristic indicated above and sequences of their best polynomial approximations in certain Banach spaces (Hardy spaces, Bergman spaces, and spaces \(B\left( {p,q,{\lambda }} \right)\) ).

### Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 250-254

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function *f*(*z*) from the Hardy space *H* _{p}, *p* ≥ 1, in disks of radii ρ, ρ_{1}, and ρ_{2}, 0 < ρ_{1} < ρ < ρ_{2} < 1.

### 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}.

### Quasiwidths and optimization of methods of mixed approximation of multidimensional singular integrals with kernels of hilbert type

Shabozov M. Sh., Vakarchuk S. B.

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 753-770

We consider the problem of application of mixed methods to the construction of algorithms, optimal in accuracy, for the calculation of multidimensional singular integrals with Hilbert-type kernels. We propose a method for the optimization of cubature formulas for singular integrals with Hilbert-type kernels based on the theory of quasiwidths.

### On exact order estimates of *N*-widths of classes of functions analytic in a simply connected domain

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 543-547

In the spaces *E* _{q}(Ω), 1 < *q* < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral *n*-widths of the classes *W* ^{r} *E* _{p}(Ω) and *W* ^{r} *E* _{p}(Ω)Ф in the case where *p* and *q* are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.

### On exact values of quasiwidths of some classes of functions

Shabozov M. Sh., Vakarchuk S. B.

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 301-308

In the Hilbert space *L* _{2}(Δ^{2}), Δ = [0, 2 π] we establish exact estimates of the Kolmogorov quasiwidths of some classes of periodic functions of two variables whose averaged modules of smoothness of mixed derivatives are majorizable by given functions.

### Exact values of widths for certain functional classes

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 133-135

For classes of functions analytic in a unit circle in a Hardy-Banach space, we obtain exact values of Kolmogorov widths in the case where the metrics of the class and the space do not coincide.

### On the best polynomial approximation of entire transcendental functions in Banach spaces. II

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1318–1322

We study the behavior of the best approximations $E_n(f)_{E′_p}$ of entire transcendental functions $f(z)$ of the order $ρ = 0$ by polynomials of at most $n$ th degree in the metric of the space $E′_p(Ω),\, p ≥ 1$. In particular, we describe the relationship between the best approximations $E_n(f)E′_p$ and the logarithmic order $ρ_L$ and type $σ_L$ of the function $f(z)$.

### On the best polynomial approximation of entire transcendental functions in banach spaces. I

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1123–1133

### Exact values of mean $n$-widths for the classes of functions analytic in the upper half plane in the Hardy space

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 814–824

In the Hardy space $H_2 ℝ_+^2$ of functions analytic in the upper half plane such that $$\sup \left\{ {\int\limits_\mathbb{R} {|f(x + iy)|^2 dx: 0< y< \infty } } \right\}< \infty ,$$ we determine mean $N$-widths and find their exact values for numerous classes of functions.

### Reconstruction of linear functionals on classes of analytic functions of two variables on the basis of generalized information

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 32–37

For classes of analytic functions defined in terms of the two-dimensional Hadamard composition, we propose a renewal method for linear functionals based on blending constructions. The best renewal methods are indicated, and the exact estimates of errors are given.

### On the diameters of certain classes of analytic functions. II

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1135–1138

In the spaces E_{q}(?), q?1, we consider the classes W^{r}E_{p}(?), p?1, consisting of analytic functions f(z) ? E_{P}(?) the integral moduli of continuity of whose r-th derivatives are majorized by a given nonnegative nondecreasing function ?.

### The best polynomial approximation of functions analytic in the unit circle

Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 838–843

### Widths of certain classes of analytic functions in the Hardy space *H*_{2}

Ukr. Mat. Zh. - 1989. - 41, № 6. - pp. 799-803

### Optimal formula for the numerical integration of a curvilinear integral of the first kind for certain classes of functions and curves

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 643–645

### Approximation by spline-curves of curves given in parametric form

Ukr. Mat. Zh. - 1983. - 35, № 3. - pp. 352 — 355