Comptes Rendus
Research article - Complex analysis and geometry
The radial limits and boundary uniqueness
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1147-1153.

The paper sheds a new light on the fundamental theorems of complex analysis due to P. Fatou, F. and M. Riesz, N. N. Lusin, I. I. Privalov, and A. Beurling. Only classical tools available at the times of Fatou are used. The proofs are very simple and in some cases – almost trivial.

L’article apporte un nouvel éclairage sur les théorèmes fondamentaux de l’analyse complexe dus à P. Fatou, F. et M. Riesz, N. N. Lusin, I. I. Privalov et A. Beurling. Seuls les outils classiques disponibles à l’époque de Fatou sont utilisés. Les preuves sont très simples et dans certains cas, presque triviales.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.609
Classification: 30J99, 30H05, 30C35
Keywords: Radial limits, Boundary uniqueness, Fatou’s Theorem, Lusin’s theorem
Mots-clés : Limites radiales, unicité frontière, théorème de Fatou, théorème de Lusin

Arthur Danielyan 1

1 Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2024__362_G10_1147_0,
     author = {Arthur Danielyan},
     title = {The radial limits and boundary uniqueness},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1147--1153},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.609},
     zbl = {07939449},
     language = {en},
}
TY  - JOUR
AU  - Arthur Danielyan
TI  - The radial limits and boundary uniqueness
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1147
EP  - 1153
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.609
LA  - en
ID  - CRMATH_2024__362_G10_1147_0
ER  - 
%0 Journal Article
%A Arthur Danielyan
%T The radial limits and boundary uniqueness
%J Comptes Rendus. Mathématique
%D 2024
%P 1147-1153
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.609
%G en
%F CRMATH_2024__362_G10_1147_0
Arthur Danielyan. The radial limits and boundary uniqueness. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1147-1153. doi : 10.5802/crmath.609. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.609/

[1] A. Beurling Ensembles exceptionnels, Acta Math., Volume 72 (1940), pp. 1-13 | DOI | MR | Zbl

[2] L. Carleson Sets of uniqueness for functions regular in the unit circle, Acta Math., Volume 87 (1952), pp. 325-345 | DOI | MR | Zbl

[3] E. F. Collingwood; A. J. Lohwater The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, 56, Cambridge University Press, 1966, xi+211 pages | DOI | MR | Zbl

[4] P. Fatou Séries trigonométriques et séries de Taylor, Acta Math., Volume 30 (1906) no. 1, pp. 335-400 | DOI | MR | Zbl

[5] K. Hoffman Banach spaces of analytic functions, Prentice Hall Series in Modern Analysis, Prentice Hall, 1962 | MR | Zbl

[6] N. N. Lusin; J. Priwaloff Sur l’unicité et la multiplicité des fonctions analytiques, Ann. Sci. Éc. Norm. Supér., Volume 42 (1925), pp. 143-191 | DOI | Numdam | MR | Zbl

[7] N. N. Lusin Sur la représentation conforme, Bull. Ivanovo-Vozn. Politech. Inst., Volume 2 (1919), pp. 77-80 | Zbl

[8] I. I. Privalov Intégrale de Cauchy, Saratov, 1919 https://archive.org/details/libgen_00292731/page/n3/mode/2up

[9] I. I. Privalov Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., 1950, 336 pages | MR

[10] M. Riesz Über die Randwerte einer analytischen Funktionen, Quatrième Congrès des Math. Scand. Stockholm (1916), pp. 27-44 | DOI | Zbl

[11] F. Riesz Über die Randwerte einer analytischen Funktionen, Math. Z., Volume 18 (1923), pp. 87-95 | DOI | MR | Zbl

[12] A. Zygmund Trigonometric series. Vol. I, Cambridge University Press, 1959, xii+383 pages | MR | Zbl

Cited by Sources:

Comments - Policy