Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp

Evgeny Abakumov; Konstantin Fedorovskiy

doi:10.1016/j.crma.2018.05.018

Complex analysis/Functional analysis

Analytic balayage of measures, Carathéodory domains, and badly approximable functions in L^p

Presented by: Gilles Pisier

Evgeny Abakumov ¹; Konstantin Fedorovskiy ^2,³

¹ Université Paris-Est, Marne-la-Vallée, France
² Bauman Moscow State Technical University, Russia
³ Saint Petersburg State University, Russia

Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 870-874.

Abstract
Résumé

We give new formulae for analytic balayage of measures supported on subsets of Carathéodory compact sets in the complex plane, and consider a related problem of description of badly approximable functions in $L^{p} (T)$ .

Nous donnons de nouvelles formules pour le balayage analytique des mesures supportées sur des sous-ensembles d'ensembles compacts de Carathéodory dans le plan complexe et considérons le problème connexe de la description des fonctions mal approximables dans $L^{p} (T)$ .

Received: 2018-04-03
Accepted: 2018-05-31
Published online: 2018-06-25

DOI: 10.1016/j.crma.2018.05.018

Author's affiliations:

Evgeny Abakumov ¹; Konstantin Fedorovskiy ^{2, 3}

¹ Université Paris-Est, Marne-la-Vallée, France
² Bauman Moscow State Technical University, Russia
³ Saint Petersburg State University, Russia

@article{CRMATH_2018__356_8_870_0,
     author = {Evgeny Abakumov and Konstantin Fedorovskiy},
     title = {Analytic balayage of measures, {Carath\'eodory} domains, and badly approximable functions in {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {870--874},
     publisher = {Elsevier},
     volume = {356},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.018},
     language = {en},
}

TY  - JOUR
AU  - Evgeny Abakumov
AU  - Konstantin Fedorovskiy
TI  - Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 870
EP  - 874
VL  - 356
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2018.05.018
LA  - en
ID  - CRMATH_2018__356_8_870_0
ER  -

%0 Journal Article
%A Evgeny Abakumov
%A Konstantin Fedorovskiy
%T Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp
%J Comptes Rendus. Mathématique
%D 2018
%P 870-874
%V 356
%N 8
%I Elsevier
%R 10.1016/j.crma.2018.05.018
%G en
%F CRMATH_2018__356_8_870_0

Evgeny Abakumov; Konstantin Fedorovskiy. Analytic balayage of measures, Carathéodory domains, and badly approximable functions in L^p. Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 870-874. doi : 10.1016/j.crma.2018.05.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.05.018/

References
Cited by

[1] A.B. Aleksandrov Badly approximable unimodular functions in weighted $L^{p}$ spaces, Comput. Methods Funct. Theory, Volume 4 (2004) no. 2, pp. 315-326

[2] A.D. Baranov; J.J. Carmona; K.Yu. Fedorovskiy Density of certain polynomial modules, J. Approx. Theory, Volume 206 (2016), pp. 1-16

[3] L. Baratchart; F.L. Nazarov; V.V. Peller Analytic approximation of matrix functions in $L^{p}$ , J. Approx. Theory, Volume 158 (2009), pp. 242-278

[4] R.V. Bessonov Analytic approximation in $L^{p}$ and coinvariant subspaces of the Hardy space, J. Approx. Theory, Volume 174 (2013), pp. 113-120

[5] J.J. Carmona; K.Yu. Fedorovskiy Conformal maps and uniform approximation by polyanalytic functions, Oper. Theory, Adv. Appl., Volume 158 (2005), pp. 109-130

[6] J.J. Carmona; K.Yu. Fedorovskiy New conditions for uniform approximation by polyanalytic polynomials, Tr. Mat. Inst. Steklova, Volume 279 (2012), pp. 227-241 (translation in Proc. Steklov Inst. Math., 279, 2012, pp. 215-229)

[7] J.J. Carmona; P.V. Paramonov; K.Yu. Fedorovskiy On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Mat. Sb., Volume 193 (2002) no. 10, pp. 75-98 (in Russian); English translation in Sb. Math., 193, 9–10, 2002, pp. 1469-1492

[8] K.M. Dyakonov Moduli and arguments of analytic functions from subspaces in $H^{p}$ that are invariant under the backward shift operator, Sib. Mat. Zh., Volume 31 (1990) no. 6, pp. 64-79 (in Russian); English translation in Sib. Math. J., 31, 6, 1990, pp. 926-939

[9] K.Yu. Fedorovskiy Carathéodory sets and analytic balayage of measures, Sb. Math. (2018) | DOI

[10] J. Garnett Bounded Analytic Function, Academic Press, 1981

[11] S.Ya. Khavinson On some extremal problems of the theory of analytic functions, Matematika, IV, Moskov. Univ. Uchen. Zap., vol. 148, Moscow State University, Moscow, 1951, pp. 133-143 English translation, Amer. Math. Soc. Transl. (2), vol. 32, 1963, American Mathematical Society, Providence, RI, USA, pp. 139-154

[12] D. Khavinson F. and M. Riesz theorem, analytic balayage, and problems in rational approximation, Constr. Approx., Volume 4 (1988), pp. 341-356

[13] D. Khavinson; J.E. McCarthy; H. Shapiro Best approximation in the mean be analytic and harmonic functions, Indiana Univ. Math. J., Volume 49 (2000) no. 4, pp. 1481-1513

[14] T. Nakazi Extremal problems in $H^{p}$ , J. Aust. Math. Soc. A, Volume 52 (1992), pp. 103-110

[15] I.I. Privalov Boundary Properties of Analytic Functions, Hochschulbücher für Mathematik, Bd. 25, GITTL, Moscow, 1950 (German transl.: Randeigenschaften analytischer Funktionen, 1956, VEB Deutscher Verlag der Wissenschaften, Berlin)

[16] W. Rudin Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987

[17] V.V. Savchuk Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions, Ukr. Math. J., Volume 59 (2007) no. 8, pp. 1163-1183

[18] J.L. Walsh Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI, USA, 1960

Cited by Sources:

Comments - Policy