arXiv subjects

Viktor V. Savchuk

Publications and source records attributed to Viktor V. Savchuk.

3 recordsLinked to original sources

arXiv2024

Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean

In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parametrized by an inner function $φ$ and a complex number $λ$, and provide explicit formula of the best approximation $e_{φ,z}(λ)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation.

math.CV↗

arXiv2022

Best Approximation-Preserving Operators over Hardy Space

Let $T_n$ be the linear Hadamard convolution operator acting over Hardy space $H^q$, $1\le q\le\infty$. We call $T_n$ a best approximation-preserving operator (BAP operator) if $T_n(e_n)=e_n$, where $e_n(z):=z^n,$ and if $\|T_n(f)\|_q\le E_n(f)_q$ for all $f\in H^q$, where $E_n(f)_q$ is the best approximation by algebraic polynomials of degree a most $n-1$ in $H^q$ space. We give necessary and sufficient conditions for $T_n$ to be a BAP operator over $H^\infty$. We apply this result to establish an exact lower bound for the best approximation of bounded holomorphic functions. In particular, we show that the Landau-type inequality $\left|\widehat f_n\right|+c\left|\widehat f_N\right|\le E_n(f)_\infty$, where $c>0$ and $n<N$, holds for every $f\in H^\infty$ iff $c\le\frac{1}{2}$ and $N\ge 2n+1$.

math.CV↗

arXiv2012

Approximation of functions of several variables by linear methods in the space $S^p$

In the spaces $S^p$ of functions of several variables, $2π$-periodic in each variable, we study the approximative properties of operators $A^\vartriangle_{\varrho,r}$ and $P^\vartriangle_{\varrho,s}$, which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes $S^pH_ω$.

math.CA↗