Weighted-$L^2$ polynomial approximation in $\mathbb {C}$
HTML articles powered by AMS MathViewer
- by Séverine Biard, John Erik Fornæss and Jujie Wu PDF
- Trans. Amer. Math. Soc. 373 (2020), 919-938 Request permission
Abstract:
We study the density of polynomials in $H^2(\Omega ,e^{-\varphi })$, the space of square integrable holomorphic functions in a bounded domain $\Omega$ in $\mathbb {C}$, where $\varphi$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $\varphi$ in a neighborhood of $\overline {\Omega }$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.References
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. MR 1805196, DOI 10.1007/978-1-4757-8137-3
- Zbigniew Błocki, Cauchy-Riemann meet Monge-Ampère, Bull. Math. Sci. 4 (2014), no. 3, 433–480. MR 3277882, DOI 10.1007/s13373-014-0058-2
- James E. Brennan, Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), no. 1, 117–168. MR 450566, DOI 10.1007/BF02386037
- T. Carleman, Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen[M], Ark. Mat. Astron. Fys., 17(9):30, 1923.
- Chen Boyong and Zhang Jinhao, On Bergman completeness and Bergman stability, Math. Ann. 318 (2000), no. 3, 517–526. MR 1800767, DOI 10.1007/s002080000105
- Jean-Pierre Demailly, Estimations $L^{2}$ pour l’opérateur $\bar \partial$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 457–511 (French). MR 690650, DOI 10.24033/asens.1434
- Harold Donnelly and Charles Fefferman, $L^{2}$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), no. 3, 593–618. MR 727705, DOI 10.2307/2006983
- O. J. Farrell, On approximation to an analytic function by polynomials, Bull. Amer. Math. Soc. 40 (1934), no. 12, 908–914. MR 1562998, DOI 10.1090/S0002-9904-1934-06002-6
- J. E. Fornæss, F. Forstneric, and E. F. Wold, Holomorphic approximation: The legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan, arXiv:1802.03924v2, 2018.
- J. E. Fornæss and J. Wu, Weighted approximation in $\mathbb {C}$, J. Math. Z. (2019), https://doi.org/10.1007/s00209-019-02321-w.
- Dieter Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). MR 604011, DOI 10.1007/978-3-0348-5812-0
- Lars Inge Hedberg, Weighted mean square approximation in plane regions, and generators of an algebra of analytic functions, Ark. Mat. 5 (1965), 541–552 (1965). MR 219729, DOI 10.1007/BF02591530
- M. Keldych, Sur l’approximation en moyenne quadratique des fonctions analytiques, Rec. Math. [Mat. Sbornik] N. S. 5 (47) (1939), 391–401 (French, with Russian summary). MR 0002591
- A. I. Markushevitch, Conformal mapping of regions with variable boundaries for the approximation of analytic functions by polynomials (Russian), PhD. thesis, 1934.
- Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
- Nessim Sibony, Approximation polynomiale pondérée dans un domaine d’holomorphie de $\textbf {C}^{n}$, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 2, x, 71–99. MR 430312
- B. A. Taylor, On weighted polynomial approximation of entire functions, Pacific J. Math. 36 (1971), 523–539. MR 284801, DOI 10.2140/pjm.1971.36.523
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- Devora Wohlgelernter, Weighted $L^{2}$ approximation of entire functions, Trans. Amer. Math. Soc. 202 (1975), 211–219. MR 355069, DOI 10.1090/S0002-9947-1975-0355069-X
Additional Information
- Séverine Biard
- Affiliation: Science Institute, University of Iceland, Dunhagi 3, IS-107 Reykjavik, Iceland
- Email: biard@hi.is
- John Erik Fornæss
- Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Alfred Getz vei 1, 7034 Trondheim, Norway
- Address at time of publication: LAMAV, Université polytechnique Hauts-de-France, Campus du Mont d’Houy, 59313 Valenciennes Cedex 9, France
- MR Author ID: 68145
- Email: severine.biard@uphf.fr
- Jujie Wu
- Affiliation: School of Mathematics and Statistics, Henan University, Kaifeng 475001, Henan Province, People’s Republic of China; and Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Alfred Getz vei 1, 7034 Trondheim, Norway
- MR Author ID: 1249622
- Email: jujie.wu@ntnu.no, 99jujiewu@tongji.edu.cn
- Received by editor(s): June 19, 2018
- Received by editor(s) in revised form: February 27, 2019
- Published electronically: October 28, 2019
- Additional Notes: The first author was supported in part by Rannis-grant 152572-051
The second author was supported by the Norwegian Research Council grant 240569
The third author was supported by the Norwegian Research Council grant 240569 and NSFC grant 11601120. The third author is the corresponding author - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 919-938
- MSC (2010): Primary 30B60, 30E10; Secondary 32A10, 32E30, 32W05, 31A05
- DOI: https://doi.org/10.1090/tran/7935
- MathSciNet review: 4068254