Kharkevych Yu. I.

UDC 517.5
We determine the subspaces of solutions of the systems of Laplace and heat-conduction differential equations isometric to the corresponding spaces of real functions determined on the set of real numbers.

For the associated sigma-functions, Jacobi theta-functions, and their logarithmic derivatives, we present asymptotic formulas valid outside an efficiently constructed exceptional sets of discs.

We obtain the asymptotic equalities for the least upper bounds of the approximations of functions from the classes $\hat{L}^{\psi}_{\beta, 1}$ by biharmonic Poisson operators in the integral metric.

We deduce asymptotic equalities for the least upper bounds of approximations of functions from the classes $W^r_{\beta} H^{\alpha}$, and $H^{\alpha}$ by biharmonic Poisson integrals in the uniform metric.

For Weierstrass functions σ(z) and ζ(z), we present the asymptotic formulas valid outside the efficiently constructed exceptional sets of discs that are much narrower than in the known asymptotic formulas.

We solve the Kolmogorov – Nikol’skii problem for biharmonic Poisson integrals on the classes of (ψ, β)- differentiable periodic functions of low smoothness in the uniform metric.

Asymptotic equalities are obtained for upper bounds of deviations of biharmonic Poisson integrals on the classes of $(\psi, \beta)$-differentiable periodic functions in the uniform metric.

We establish the Nevanlinna characteristics of the Weierstrass zeta function and show that none of the values $a \in \overline{C}$ is exceptional in the Nevanlinna sense for this function.

We obtain asymptotic equalities for upper bounds of approximations of functions from the class $C_{β,∞} ψ$ by Poisson integrals in the metric of the space $C$.

We determine the exact values of upper bounds of approximations by biharmonic Poisson integrals on classes of conjugate differentiable functions in uniform and integral metrics.

We obtain the exact values of upper bounds of approximations of classes of periodic conjugate differentiable functions by their Abel–Poisson integrals in uniform and integral metrics.

We obtain asymptotic equalities for upper bounds of approximations of functions from the class $\hat{C}^{\psi}_{\beta, \infty}$ by the Poisson biharmonic operators in the uniform metric.

Complete asymptotic decompositions are obtained for values of exact upper bounds of approximations of functions from the classes $W^r_1,\quad r \in N,$ and WJr, $\overline{W}^r_1,\quad r \in N\backslash\{1\}$, by their biharmonic Poisson integrals.

Asymptotic equalities are obtained for upper bounds of approximations of functions from the classes $C^{\psi}_{\beta \infty}$ and $L^{\psi}_{\beta 1}$ by the Weierstrass integrals.

In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m-and ((N ? m))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality.

We obtain asymptotic equalities for upper bounds of approximations of functions on the classes \(\hat C_{\beta ,\infty }^\psi\) and \(\hat L_{\beta ,1}^\psi\) by Abel-Poisson operators.

We obtain asymptotic equalities for upper bounds of the deviations of operators generated by λ-methods (defined by a collection Λ={λ_σ(·)} of functions continuous on [0; ∞) and depending on a real parameter σ) on classes of (ψ, β)-differentiable functions defined on the real axis.

We determine the exact values and asymptotic decompositions of upper bounds of approximations by biharmonic Poisson integrals on classes of periodic differentiable functions.

On a class of differentiable functions W ^r and the class \(\overline W ^r \) of functions conjugate to them, we obtain a complete asymptotic expansion of the upper bounds \(\mathcal{E}(\mathfrak{N},A\rho )_C \) of deviations of the harmonic Poisson integrals of the functions considered.

We determine the exact value of the upper bound for the deviation of the triharmonic Poisson integral from functions of the Hölder class.

We determine the exact value of the upper bound of the deviation of biharmonic Poisson integrals from functions of the Hölder class.