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Mikhail A. Komarov

Publications and source records attributed to Mikhail A. Komarov.

6 recordsLinked to original sources
arXiv2025

On the precise form of the inverse Markov factor for convex sets

Let $K\subset \mathbb{C}$ be a convex compact set, and let $Π_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Turán type inverse Markov factor is defined by $M_n(K)=\inf_{P\in Π_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound $M_n(K)\ge c\left(wn/d^2+\sqrt{n}/d\right)$, $c:=0.00015$, where $d>0$ is the diameter of $K$ and $w\ge 0$ is the minimal width (the smallest distance between two parallel lines between which $K$ lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28\left(wn/d^2+\sqrt{n}/d\right)$ for all $n,w,d$.

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arXiv2024

On reverse Markov-Nikol'skii inequalities for polynomials with restricted zeros

Let $Π_n$ be the class of algebraic polynomials $P$ of degree $n$, all of whose zeros lie on the segment $[-1,1]$. In 1995, S.P. Zhou has proved the following Turán type reverse Markov-Nikol'skii inequality: $\|P'\|_{L_p[-1,1]}>c\, {(\sqrt{n})}^{1-1/p+1/q}\, \|P\|_{L_q[-1,1]}$, $P\in Π_n$, where $0 0$ is a constant independent of $P$ and $n$). We show that Zhou's estimate remains true in the case $p=\infty$, $q>1$. Some of related Turán type inequalities are also discussed.

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arXiv2022

A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle

Let $g_n$, $n=1,2,\dots$, be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $g_n(z)=(z-z_{1})^{-1}+\dots+(z-z_{n})^{-1}$, $|z_1|=\dots=|z_n|=1$. For any $p>0$, we establish the bound \[\int_{-1}^1 |g_n(x)|^p\, dx>C_p\, n^{p-1},\] sharp in the order of the quantity $n$, where $C_p>0$ is a constant, depending only on $p$. The particular case $p=1$ of this inequality can be considered as a stronger variant of the well-known estimate $\iint_{|z|<1} |g_n(z)|\,dxdy>c>0$ for the area integral of $g_n$, obtained by D.J. Newman (1972). The result also shows that the set $\{g_n\}$ is not dense in the spaces $L_p[-1,1]$, $p\ge 1$.

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arXiv2019

Rate of approximation of $zf'(z)$ by special sums associated with the zeros of the Bessel polynomials

Let $α_{n1},\dots,α_{nn}$ be the zeros of the $n$th Bessel polynomial $y_n(z)$ and let $a_{nk}=1-α_{nk}/2$, $b_{nk}=1+α_{nk}/2$ $(k=1,\dots,n)$. We propose the new formula \[z f'(z)\approx \sum_{k=1}^n \big(f(a_{nk} z)-f(b_{nk} z)\big)\] for numerical differentiation of analytic functions $f(z)=\sum_0^\infty f_m z^m$. This formula is exact for all polynomials of degree at most $2n$. We find the sharp order of nonlocal estimate of the corresponding remainder for the case when all $|f_m|\le 1$. The estimate shows a high rate of convergence of the differentiating sums to $zf'(z)$ on compact subsets of the open unit disk, namely, $O(0.85^n n^{1-n})$ as $n\to \infty$.

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arXiv2018

Caratheodory type representation with unit weights and related approximation problems

For arbitrary $n$ complex numbers $a_{ν-1}$, $ν=1,\dots,n$, where $n$ is sufficiently large, we get the representation in the form of power sums: $a_{ν-1}=λ_1^ν+\dots+λ_{2n+1}^ν$, where $λ_k$ are distinct points, such that $|λ_k|=1$. We study several applications to the problem of approximation by exponential sums and by $h$-sums, to the problem of extracting of harmonics from trigonometric polynomials. The result is based on an estimate for the uniform approximation rate of bounded analytic in the unit disk functions by logarithmic derivatives of polynomials, all of whose zeros lie on the unit circle $C : |z| = 1$. Our result is a modification of classical Carathéodory representation $a_{ν-1}=\sum_{k=1}^{n} X_k λ_k^ν$, $ν=1,2,\dots,n$, where weights $X_k\ge 0$, and $λ_k$ are distinct points, such that $|λ_k|=1$.

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