# Savchuk V. V.

### Best approximations of the Cauchy – Szegö kernel in the mean on the unit circle

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 708-714

We compute the values of the best approximations of the Cauchy – Szeg¨o kernel in the mean on the unit circle by quasipolynomials with respect to the Takenaka – Malmquist system.

### Application of the Faber polynomials in proving combinatorial identities

Abdullayev F. G., Imash kyzy M., Savchuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 151-164

We study the possibility of application of the Faber polynomials in proving some combinatorial identities. It is shown that the coefficients of Faber polynomials of mutually inverse conformal mappings generate a pair of mutually invertible relations. We prove two identities relating the coefficients of Faber polynomials and the coefficients of Laurent expansions of the corresponding conformal mappings. Some examples are presented.

### Oleksandr Ivanovych Stepanets’ (on his 75th birthday)

Romanyuk A. S., Romanyuk V. S., Samoilenko A. M., Savchuk V. V., Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 579

### Direct and inverse theorems on the approximation of 2π -periodic functions by Taylor – Abel – Poisson operators

Prestin J., Savchuk V. V., Shydlich A. L.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 657

We obtain direct and inverse theorems on the approximation of 2\pi -periodic functions by Taylor – Abel – Poisson operators in the integral metric.

### Exact constants in inequalities for the Taylor coefficients of bounded holomorphic functions in a polydisc

Meremelya I. Yu., Savchuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1690-1697

We determine the exact constants $L_{m,n}(X)$ in the inequalities of the form $|\hat f(m)|\leq L_{m,n}(X)(1 − |\hat f(n)|)$ for the pairs of Taylor coefficients $\hat f(m)$ and $\hat f(n)$ on some classes $X$ of bounded holomorphic functions in a polydisc.

### Best Approximations for the Cauchy Kernel on the Real Axis

Chaichenko S. O., Savchuk V. V.

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1540–1549

We compute the values of the best approximations for the Cauchy kernel on the real axis $ℝ$ by some subspaces from $L_q (ℝ)$. This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka–Malmquist system from the functions lying in the unit ball of the Hardy space $H_p,\; 2 ≤ p < ∞$.

### Approximation of holomorphic functions of Zygmund class by Fejer means

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1148-1152

We obtain an asymptotic equality for the upper bounds of deviations of Fejer means on the Zygmund class of functions holomorphic in the unit disk.

### Summation of *p*-Faber series by the Abel–poisson method in the integral metric

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 660–673

We establish conditions on the boundary \( \Gamma \) of a bounded simply connected domain \( \Omega \subset \mathbb{C} \) under which the *p*-Faber series of an arbitrary function from the Smirnov space \( {E_p}\left( \Omega \right),1 \leqslant p < \infty \), can be summed by the Abel–Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space \( {L_p}\left( \Gamma \right) \).

### Letter to the editor

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 289

### Linear methods for approximation of some classes of holomorphic functions from the Bergman space

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 783–795

We construct a linear method of the approximation $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ in the unit disk of classes of holomorphic functions $A^{\psi}_p$ that are the Hadamard convolutions of unit balls of the Bergman space $A_p$ with reproducing kernels $\psi(z) = \sum^\infty_{k=0}\psi_k z^k.$ We give conditions on $\psi$ under which the method $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ approximate the class $A^{\psi}_p$ in metrics of the Hardy space $H_s$ and Bergman space $A_s,\; 1 \leq s \leq p,$ with error that coincides in order with a value of the best approximation by algebraic polynomials.

### Best linear methods of approximation and optimal orthonormal systems of the Hardy space

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 636–646

We construct the best linear methods of the approximation of functions from the Hardy space *H _{p }* on compact subsets of the unit disk. We show that the Takenaka - Malmquist systems are such systems of
functions that are orthonormal on the unit circle and optimal for the construction of the best linear methods of approximation.

### Approximation of holomorphic functions by Taylor-Abel-Poisson means

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1253–1260

We investigate approximations of functions $f$ holomorphic in the unit disk by means $A_{\rho, r}(f)$ for $\rho \rightarrow 1_-$.
In terms of an error of the approximation by these means, the constructive characteristic of classes of holomorphic functions $H_p^r \text{\;Lip\,}\alpha$ is given.
The problem of the saturation of $A_{\rho, r}(f)$ in the Hardy space $H_p$ is solved.

### Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1047–1067

We find necessary and sufficient conditions under which a real function from $L_p(\mathbb{T}),\; 1 \leq p < \infty$, is badly approximable by the Hardy subspace $H_p^0: = \{f \in H_p:\; F(0) = 0\}$. In a number of cases, we obtain exact values for the best approximations in the mean of functions holomorphic in the unit disk by functions that are holomorphic outside the unit disk. We use obtained results in determining exact values of the best polynomial approximations and га-widths of some classes of holomorphic functions. We find necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.

### International Conference "Mathematical Analysis and Differential Equations and Applications"

Samoilenko A. M., Savchuk V. V., Sokolenko I. V., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 431

### Best linear methods for the approximation of functions of the Bergman class by algebraic polynomials

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1674–1685

On concentric circles $T_{ϱ} = {z ∈ ℂ: ∣z∣ = ϱ},\; 0 ≤ ϱ < 1$, we determine the exact values of the quantities of the best approximation of holomorphic functions of the Bergman class $A_p, 2 ≤ p ≤ ∞$, in the uniform metric by algebraic polynomials generated by linear methods of summation of Taylor series. For $1 ≤ p < 2$, we establish exact order estimates for these quantities.

### Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 522–542

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

### Best approximation of reproducing kernels of spaces of analytic functions

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 947–959

We obtain exact values for the best approximation of a reproducing kernel of a system of *p*-Faber polynomials by functions of the Hardy space *H* _{q}, *p* ^{-1} + *q* ^{-1} = 1 and a Szegö reproducing kernel of the space *E* ^{2}(Ω) in a simply connected domain Ω with rectifiable boundary.

### Best Linear Methods of Approximation of Functions of the Hardy Class $H_p$

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 919-925

We determine the exact value of the best linear polynomial approximation of a unit ball of the Hardy space $H_p, 1 ≤ p ≤ ∞$, on concentric circles $Tρ = z ∈ C:|z|=ρ, 0 ≤ ρ < 1$, in the uniform metric. We construct the best linear method of approximation and prove the uniqueness of this method.

### Norms of Multipliers and Best Approximations of Holomorphic Functions of Many Variables

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1669-1680

We show that the Lebesgue–Landau constants of linear methods for summation of Taylor series of functions holomorphic in a polydisk and in the unit ball from \(\mathbb{C}^m\) over triangular domains do not depend on the number *m*. On the basis of this fact, we find a relation between the complete and partial best approximations of holomorphic functions in a polydisk and in the unit ball from \(\mathbb{C}^m\) .

### Approximation of Cauchy-Type Integrals

Savchuk V. V., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 706-740

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of ψ-integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded *r*th derivatives (here, *r* is a natural number).

On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of ψ-integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.

### On the mean-value theorem for analytic functions

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1143–1147

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved.