Geodesic
Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk
Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 3-10.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a function $h$ analytic in the unit disk $D$, we study the density in the space $A(D)$ of functions analytic inside $D$ of the set $S(h,E)$ of sums of the form $\sum_k\lambda_kh(\lambda_kz)$ with parameters $\lambda_k\in E$, where $E$ is a compact subset of $\overline D$. It is proved, in particular, that if the compact set $E$ “surrounds” the point $0$ and all Taylor coefficients of the function $h$ are nonzero, then $S(h,E)$ is dense in $A(D)$.
Keywords: approximation, analytic function, density, $h$-sum. 
@article{MZM_2018_104_1_a0,
     author = {P. A. Borodin},
     title = {Approximation by {Sums} of the {Form} $\sum_k\lambda_kh(\lambda_kz)$ in the {Disk}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {3--10},
     publisher = {mathdoc},
     volume = {104},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a0/}
}
TY  - JOUR
AU  - P. A. Borodin
TI  - Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk
JO  - Matematičeskie zametki
PY  - 2018
SP  - 3
EP  - 10
VL  - 104
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a0/
LA  - ru
ID  - MZM_2018_104_1_a0
ER  - 
%0 Journal Article
%A P. A. Borodin
%T Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk
%J Matematičeskie zametki
%D 2018
%P 3-10
%V 104
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a0/
%G ru
%F MZM_2018_104_1_a0
P. A. Borodin. Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk. Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 3-10. http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a0/

[1] V. I. Danchenko, “Ob approksimativnykh svoistvakh summ vida $\sum_k\lambda_kh(\lambda_k z)$”, Matem. zametki, 83:5 (2008), 643–649 | DOI | MR | Zbl

[2] V. I. Danchenko, D. Ya. Danchenko, “O priblizhenii naiprosteishimi drobyami”, Matem. zametki, 70:4 (2001), 553–559 | DOI | MR | Zbl

[3] O. N. Kosukhin, “Ob approksimatsionnykh svoistvakh naiprosteishikh drobei”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2001, no. 4, 54–59 | MR | Zbl

[4] P. A. Borodin, O. N. Kosukhin, “O priblizhenii naiprosteishimi drobyami na deistvitelnoi osi”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2005, no. 1, 3–8 | MR | Zbl

[5] P. Chunaev, “Least deviation of logarithmic derivatives of algebraic polynomials from zero”, J. Approx. Theory, 185 (2014), 98–106 | DOI | MR | Zbl

[6] M. A. Komarov, “Kriterii nailuchshego ravnomernogo priblizheniya naiprosteishimi drobyami v terminakh alternansa. II”, Izv. RAN. Ser. matem., 81:3 (2017), 109–133 | DOI | MR | Zbl

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

[8] P. V. Chunaev, “Ob ekstrapolyatsii analiticheskikh funktsii summami vida $\sum_k\lambda_k h(\lambda_k z)$”, Matem. zametki, 92:5 (2012), 794–797 | DOI | MR | Zbl

[9] P. Chunaev, V. Danchenko, “Approximation by amplitude and frequency operators”, J. Approx. Theory, 207 (2016), 1–31 | DOI | MR | Zbl

[10] P. A. Borodin, “Priblizhenie summami sdvigov odnoi funktsii na okruzhnosti”, Izv. RAN. Ser. matem., 81:6 (2017), 23–37 | DOI | Zbl

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

[12] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, GITTL, M.–L., 1950 | MR | Zbl

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