Approximation of holomorphic Legendrian curves with jet-interpolation

Abstract

V članku dokažemo več rezultatov o interpolaciji holomorfnih Legendrovih krivulj v liho razsežnih kompleksnih evklidskih prostorih. V posebnem dokažemo, da poljubna števna podmnožica v ▫$\mathbb{C}^{2n+1}$▫ leži na injektivno imerzirani izotropni ploskvi s predpisano kompleksno strukturo. Če podmnožica nima stekališč lahko dosežemo, da je ploskev pravilno vložena. Za holomorfne Legendrove krivulje dokažemo tudi izrek o aproksimaciji v smislu Carlemana z interpolacijo. Natančneje pokažemo, da je Legendrovo krivuljo, ki je definirana na določeni vrsti zaprte množice v odprti Riemannovi ploskvi ▫$\mathcal{R}$▫, možno aproksimirati v ▫$\mathcal{C}^0(\mathcal{R})$▫-topologiji s celo Legendrovo krivuljo, ki ji določimo Taylorjeve polinome poljubne končne stopnje na neki zaprti diskretni podmnožici v ▫$\mathcal{R}$▫. Pokažemo, da je tovrstna aproksimacija v posebnih primerih možna s pravilnimi vložitvami.

Keywords

holomorfne Legendrove krivulje;Carlemanova aproksimacija;Mergelyanova aproksimacija;holomorphic Legendrian curve;Carleman approximation;Mergelyan approximation;

Data

Language:	English
Year of publishing:	2024
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.5
COBISS:	173063683
ISSN:	0022-247X
Views:	38
Downloads:	5
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Aproksimacija holomorfnih Legendrovih krivulj z interpolacijo brstičev
Secondary abstract:	We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in ▫$\mathbb{C}^{2n+1}$▫ lies on an injectively immersed isotropic surface with a prescribed complex structure. If the set has no accumulation points, the surface may be taken properly embedded. We also prove a Carleman-type theorem for holomorphic Legendrian curves with interpolation. Namely, a Legendrian curve, defined on a certain type of unbounded closed set in a given open Riemann surface ▫$\mathcal{R}$▫, may be approximated in the ▫$\mathcal{C}^0$▫-topology by an entire Legendrian curve with prescribed finite-order Taylor polynomials at a closed discrete set of points in ▫$\mathcal{R}$▫. Under suitable conditions, the approximating map may be made into a proper embedding.
Secondary keywords:	holomorfne Legendrove krivulje;Carlemanova aproksimacija;Mergelyanova aproksimacija;
Type (COBISS):	Article
Pages:	25 str.
Volume:	ǂVol. ǂ531
Issue:	ǂiss. ǂ2, part 1, [article no.] 127839
Chronology:	Mar. 2024
DOI:	10.1016/j.jmaa.2023.127839
ID:	21708535