Geodesic
S. R. Nasyrov's Problem of Approximation by Simple Partial Fractions on an Interval
Matematičeskie zametki, Tome 115 (2024) no. 4, pp. 568-577 Cet article a éte moissonné depuis la source Math-Net.Ru

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In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space $L_2[-1,1]$. In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov's problem different from Komarov's one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on $[-1,1]$; (b) of the density in $L_2[-1,1]$ of simple partial fractions with poles on the boundary of a given domain for which $[-1,1]$ is an inner chord.
Keywords: approximation, simple partial fraction
Mots-clés : Lebesgue space, constraints on poles. 
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P. A. Borodin; A. M. Ershov. S. R. Nasyrov's Problem of Approximation by Simple Partial Fractions on an Interval. Matematičeskie zametki, Tome 115 (2024) no. 4, pp. 568-577. http://geodesic.mathdoc.fr/item/MZM_2024_115_4_a6/

[1] V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49 | MR

[2] J. Korevaar, “Asymptotically neutral distributions of electrons and polynomial approximation”, Ann. of Math. (2), 80:2 (1964), 403–410 | DOI | MR

[3] P. A. Borodin, “Priblizhenie naiprosteishimi drobyami s ogranicheniem na polyusy. II”, Matem. sb., 207:3 (2016), 19–30 | DOI | MR

[4] J. M. Elkins, “Approximation by polynomials with restricted zeros”, J. Math. Anal. Appl., 25:2 (1969), 321–336 | DOI | MR

[5] P. A. Borodin, K. S. Shklyaev, “Priblizhenie naiprosteishimi drobyami v neogranichennykh oblastyakh”, Matem. sb., 212:4 (2021), 3–28 | DOI | MR

[6] M. A. Komarov, “A lower bound for the $L_2[-1, 1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle”, Probl. Anal. Issues Anal., 8:2 (2019), 67–72 | DOI | MR

[7] M. A. Komarov, “A Newman type bound for $L_p[-1, 1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle”, Constr. Approx., 58:3 (2023), 551–563 | DOI | MR

[8] M. A. Komarov, “Plotnost naiprosteishikh drobei s polyusami na okruzhnosti v vesovykh prostranstvakh dlya kruga i otrezka”, Vestnik SPbGU. Matematika. Mekhanika. Astronomiya, 2023 (to appear)

[9] P. A. Borodin, “Plotnost polugruppy v banakhovom prostranstve”, Izv. RAN. Ser. matem., 78:6 (2014), 21–48 | DOI | MR | Zbl

[10] Dzh. Distel, Geometriya banakhovykh prostranstv, Vischa shkola, Kiev, 1980 | MR

[11] T. Gamelin, Ravnomernye algebry, Mir, M., 1973 | MR