Geodesic
On a~problem of Beurling
Algebra i analiz, Tome 30 (2018) no. 1, pp. 1-19.

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

This article is an extended version of a lecture presented to a conference at Brown University on 11 June 2017 in celebration of John Wermer's 90th birthday. Here, we discuss a complete solution to the weighted approximation problem for polynomials on an arbitrary bounded simply connected domain $\Omega$ in the complex plane. The problem had been studied extensively by Keldysh [13] prior to 1941 in the context of weighted $L^2$-approximation, and more recently by Beurling [5], where the emphasis is on uniform approximation.
Keywords: weighted polynomial approximation, quasi-analyticity, asymptotically holomorphic functions, harmonic measure. 
@article{AA_2018_30_1_a0,
     author = {J. E. Brennan},
     title = {On a~problem of {Beurling}},
     journal = {Algebra i analiz},
     pages = {1--19},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2018_30_1_a0/}
}
TY  - JOUR
AU  - J. E. Brennan
TI  - On a~problem of Beurling
JO  - Algebra i analiz
PY  - 2018
SP  - 1
EP  - 19
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2018_30_1_a0/
LA  - en
ID  - AA_2018_30_1_a0
ER  - 
%0 Journal Article
%A J. E. Brennan
%T On a~problem of Beurling
%J Algebra i analiz
%D 2018
%P 1-19
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2018_30_1_a0/
%G en
%F AA_2018_30_1_a0
J. E. Brennan. On a~problem of Beurling. Algebra i analiz, Tome 30 (2018) no. 1, pp. 1-19. http://geodesic.mathdoc.fr/item/AA_2018_30_1_a0/

[1] Akhiezer N. I., “O vzveshennom priblizhenii nepreryvnykh funktsii mnogochlenami na vsei chislovoi osi”, Uspekhi mat. nauk, 11:4 (1956), 3–43 | MR | Zbl

[2] Bernstein S. N., “Le problème de l'approximation des fonctions continues sur l'axe réel et l'une des ses applications”, Bull. Soc. Math. France, 52 (1924), 399–410 | DOI | MR

[3] Beurling A., On quasianalyticity and general distributions, Multilithed Lecture Notes, Summer School, Stanford Univ., Stanford, CA, 1961

[4] Beurling A., “Analytic continuation across a linear boundary”, Acta Math., 128:3–4 (1972), 153–182 | DOI | MR

[5] Beurling A., The collected works of Arne Beurling, v. I, Complex analysis, Birkhäuser, Boston, MA, 1989 | MR

[6] Brennan J. E., “Point evaluations, invariant subspaces, and approximation in the mean by polynomials”, J. Funct. Anal., 34:3 (1979), 407–420 | DOI | MR

[7] Brennan J. E., “Functions with rapidly decreasing negative Fourier coefficients”, Lecture Notes in Math., 1275, Springer, Berlin, 1987, 31–43 | DOI | MR

[8] Brennan J. E., “Weighted polynomial approximation and quasianalyticity for general sets”, Algebra i analiz, 6:4 (1994), 69–89 | MR | Zbl

[9] Brennan J. E., “Three problems in function theory”, Oper. Theory. Adv. Appl., 261, Birkhäuser, Basel, 2017, 191–227 | DOI

[10] Dynkin E. M., “Funktsii s dannoi otsenkoi $\partial f/\partial\bar z$ i teorema N. Levinsona”, Mat. sb., 88(131):2 (1972), 182–190 | MR | Zbl

[11] Garnett J. B., Marshall D. E., Harmonic measure, New Math. Monogr., 2, Cambridge Univ. Press, Cambridge, 2005 | MR

[12] Khavin V. P., “Approksimatsiya analiticheskimi funktsiyami v srednem”, Dokl. AN SSSR, 178:5 (1968), 1025–1028 | MR | Zbl

[13] Keldysh M. V., “O zamknutosti ortogonalnykh s vesom sistem polinomov”, Dokl. AN SSSR, 30:9 (1941), 771–773

[14] Keldysh M. V., “Sur l'approximation en moyenne par polynomes des fonctions d'une variable complexe”, Mat. sb., 16(58):1 (1945), 1–20 | MR | Zbl

[15] Koosis P., The logarithmic integral, v. I, Cambridge Stud. Adv. Math., 12, Cambridge Univ. Press, Cambridge, 1988 ; The logarithmic integral, v. II, Cambridge Stud. Adv. Math., 21, Cambridge Univ. Press, Cambridge, 1992 | MR | MR

[16] Levinson N., “On a class of non-vanishing functions”, Proc. London Math. Soc. (2), 41:5 (1936), 393–407 | DOI | MR

[17] Mergelyan S. N., “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, Uspekhi mat. nauk, 7:2 (1952), 31–122 | MR | Zbl

[18] Mergelyan S. N, “O polnote sistem analiticheskikh funktsii”, Uspekhi mat. nauk, 8:4 (1953), 3–63 | MR | Zbl

[19] Mergelyan S. N., “Vesovye priblizheniya mnogochlenami”, Uspekhi mat. nauk, 11:5 (1956), 107–152 | MR | Zbl

[20] Pommerenke Ch., Boundary behavior of conformal maps, Grundlehren Math. Wiss., 299, Springer-Verlag, Berlin, 1992 | DOI | MR

[21] Sinanyan S. O., “Approksimatsiya analiticheskimi funktsiyami i polinomami v srednem po ploschadi”, Mat. sb., 69(111):4 (1966), 546–578 | MR | Zbl

[22] Volberg A. L., “Logarifm pochti analiticheskoi funktsii summiruem”, Dokl. AN SSSR, 265:6 (1982), 1297–1302 | MR

[23] Volberg A. L., Erikke B., “Summiruemost logarifma pochti analiticheskoi funktsii i obobschenie teoremy Levinsona–Kartrait”, Mat. sb., 130(172):3 (1986), 335–348 | MR | Zbl

[24] Wermer J., Seminar über Funktionen-Algebren, Lecture Notes in Math., 1, Springer-Verlag, Berlin, 1964 | DOI | MR