Geodesic
Approximation by simple partial fractions with constraints on the poles. II
Sbornik. Mathematics, Tome 207 (2016) no. 3, pp. 331-341 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is shown that if a compact set $K$ not separating the plane $\mathbb C$ lies in the union $\widehat{E}\setminus E$ of the bounded components of the complement of another compact set $E$, then the simple partial fractions (the logarithmic derivatives of polynomials) with poles in $E$ are dense in the space $AC(K)$ of functions that are continuous on $K$ and analytic in its interior. It is also shown that if a compact set $K$ with connected complement lies in the complement $\mathbb C\setminus\overline{D}$ of the closure of a doubly connected domain $D\subset \overline{\mathbb C}$ with bounded connected components of the boundary $E^+$ and $E^-$, then the differences $r_1-r_2$ of the simple partial fractions such that $r_1$ has its poles in $E^+$ and $r_2$ has its poles in $E^-$ are dense in the space $AC(K)$. Bibliography: 9 titles.
Keywords: uniform approximation, restriction on the poles, neutral distribution
Mots-clés : simple partial fractions, condenser. 
@article{SM_2016_207_3_a1,
     author = {P. A. Borodin},
     title = {Approximation by simple partial fractions with constraints on the {poles.~II}},
     journal = {Sbornik. Mathematics},
     pages = {331--341},
     year = {2016},
     volume = {207},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_3_a1/}
}
TY  - JOUR
AU  - P. A. Borodin
TI  - Approximation by simple partial fractions with constraints on the poles. II
JO  - Sbornik. Mathematics
PY  - 2016
SP  - 331
EP  - 341
VL  - 207
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2016_207_3_a1/
LA  - en
ID  - SM_2016_207_3_a1
ER  - 
%0 Journal Article
%A P. A. Borodin
%T Approximation by simple partial fractions with constraints on the poles. II
%J Sbornik. Mathematics
%D 2016
%P 331-341
%V 207
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2016_207_3_a1/
%G en
%F SM_2016_207_3_a1
P. A. Borodin. Approximation by simple partial fractions with constraints on the poles. II. Sbornik. Mathematics, Tome 207 (2016) no. 3, pp. 331-341. http://geodesic.mathdoc.fr/item/SM_2016_207_3_a1/

[1] P. A. Borodin, “Approximation by simple partial fractions with constraints on the poles”, Sb. Math., 203:11 (2012), 1553–1570 | DOI | DOI | MR | Zbl

[2] V. I. Danchenko, D. Ya. Danchenko, “Approximation by simplest fractions”, Math. Notes, 70:4 (2001), 502–507 | DOI | DOI | MR | Zbl

[3] J. Korevaar, “Asymptotically neutral distributions of electrons and polynomial approximation”, Ann. of Math. (2), 80:3 (1964), 403–410 | DOI | MR | Zbl

[4] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, R.I., 1969, vi+676 pp. | MR | MR | Zbl | Zbl

[5] I. I. Priwalow, Randeigenschaften analytischer Funktionen, 25, 2. Aufl., Hochschulbücher für Mathematik, Berlin, 1956, viii+247 pp. | MR | MR | Zbl | Zbl

[6] M. A. Lawrentjew, B. W. Schabat, Methoden der komplexen Funktionentheorie, Math. Naturwiss. Tech., 13, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967, x+846 pp. | MR | MR | Zbl | Zbl

[7] T. W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969, xiii+257 pp. | MR | Zbl | Zbl

[8] P. A. Borodin, “Density of a semigroup in a Banach space”, Izv. Math., 78:6 (2014), 1079–1104 | DOI | DOI | MR | Zbl

[9] Ch. K. Chui, “On approximation in the Bers spaces”, Proc. Amer. Math. Soc., 40:2 (1973), 438–442 | DOI | MR | Zbl