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

Komarov, M. A.

doi:10.1007/s00365-023-09622-8

Cited by 5 publications

(2 citation statements)

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“…In 2014 Nasyrov posed the following problem: are the simple partial fractions with poles on the unit circle dense in the complex space L 2 [−1, 1]? This problem was fixed in [10] and was solved in the negative by Komarov [51], who considerably generalized and clarified his result subsequently (see [55]). Theorem 3.17 (Komarov [51], [55]).…”

Section: Various Function Spacesmentioning

confidence: 99%

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Density of quantized approximations

Borodin,

Shklyaev

2023

Russian Math. Surveys

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The paper contains a review of known results and proofs of new results on conditions on a set $M$ in a Banach space $X$ that are necessary or sufficient for the additive semigroup $R(M)=\{x_1+…+x_n\colon x_k\in M, n\in {\mathbb N}\}$ to be dense in $X$. We prove, in particular, that if $M$ is a rectifiable curve in a uniformly smooth real space $X$, and $M$ does not lie entirely in any closed half-space, then $R(M)$ is dense in $X$. We present known and new results on the approximation by simple partial fractions (logarithmic derivatives of polynomials) in various spaces of functions of a complex variable. Meanwhile, some well-known theorems, in particular, Korevaar's theorem, are derived from new general results on the density of a semigroup. We also study approximation by sums of shifts of one function, which are a natural generalization of simple partial fractions. Bibliography: 79 titles.

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Section: Various Function Spacesmentioning

confidence: 99%

“…This problem was fixed in [10] and was solved in the negative by Komarov [51], who considerably generalized and clarified his result subsequently (see [55]). Theorem 3.17 (Komarov [51], [55]). The simple partial fractions SF(C) with poles on the unit circle C are not dense in the complex space L p [−1, 1] for p ⩾ 1.…”

Section: Various Function Spacesmentioning

confidence: 99%