×

zbMATH — the first resource for mathematics

Orthogonal measures on the boundary of a Riemann surface and polynomial hull of compacts of finite length. (English) Zbl 0951.32005
Summary: Let \(\mu\) be an orthogonal measure with compact support of finite length in \(\mathbb{C}^n\). The author proves, under a very weak hypothesis of regularity on the support \(\text{(Supp} \mu)\) of \(\mu\), that this measure is characterized by its boundary values (in the weak sense of currents) of the current \([T]\wedge\varphi\), where \(T\) is an analytic subset of dimension 1 of \(\mathbb{C}^n \setminus\text{Supp} \mu\) and \(\varphi\) is a holomorphic \((1,0)\)-form on \(T\). This allows to prove that the polynomial hull \(\widehat X\) of a compactum \(X\subset\mathbb{C}^n\) of finite length with a weak regularity assumption is its union with an analytic subset of pure dimension 1 of \(\mathbb{C}^n \setminus X\). He also proves that the measure \(\mu\) can be decomposed into a sum of orthogonal measures with small support. He deduces that a continuous function on \(\widehat X\) is approximable by polynomials if and only if it is locally approximable.

MSC:
32C30 Integration on analytic sets and spaces, currents
32F17 Other notions of convexity in relation to several complex variables
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexander, H., Polynomial approximation and hulls in sets of finite linear measure in \(C\)^{n}, Amer. J. math., 93, 65-74, (1971) · Zbl 0221.32011
[2] Alexander, H., The polynomial hull of a set of finite linear measure in \(C\)^{n}, J. anal. math., 47, 238-242, (1986) · Zbl 0615.32009
[3] Alexander, H., Ends of varieties, Bull. soc. math. France, 120, 523-547, (1992) · Zbl 0783.32004
[4] Bishop, E., Subalgebras of functions on a Riemann surface, Pacific J. math., 8, 29-50, (1958) · Zbl 0083.06201
[5] Bishop, E., Analyticity in certain Banach algebras, Trans. amer. math. soc., 102, 507-544, (1962) · Zbl 0112.07301
[6] Dinh, T.C., Chaı̂nes holomorphes à bord rectifiable, C.R. acad. sci. Paris Sér. I. math., 322, 1135-1140, (1996) · Zbl 0865.32007
[7] Dinh, T.C., Enveloppe polynomiale d’un compact de longueur finie et chaı̂nes holomorphes à bord rectifiable, Acta math., 180, 31-67, (1998) · Zbl 0942.32008
[8] Dolbeault, P.; Henkin, G., Surfaces de Riemann de bord donné dans \(C\)\(P\)^{n}, Aspects math., 26, (1994), Vieweg Wiesbaden, p. 163-187
[9] Dolbeault, P.; Henkin, G., Chaı̂nes holomorphes de bord donné dans \(C\)\(P\)^{n}, Bull. soc. math. France, 125, 383-445, (1997) · Zbl 0942.32007
[10] Federer, F., Geometric measure theory, Grundlenhren der math. wiss., 285, (1988), Springer-Verlag Berlin/New York
[11] Gamelin, T.W., Uniform algebras, (1969), Prentice Hall New York · Zbl 0213.40401
[12] Gamelin, T.W., Polynomial approximation on thin sets, Symposium on several complex variables, park city, Utah, 1970, Lecture notes in math., 184, (1971), Springer-Verlag New York, p. 50-78 · Zbl 0216.10502
[13] Goluzin, G.M., Geometric theory of functions of a complex variable, Trans. math. monogr., 26, (1969), Amer. Math. Soc Providence · Zbl 0183.07502
[14] Harvey, R.; Lawson, B., On boundaries of complex analytic varieties, I, Ann. of math., 102, 233-290, (1975) · Zbl 0317.32017
[15] Henkin, G., The abel – radon transform and several complex variables, Ann. of math. stud., 137, 223-275, (1995) · Zbl 0848.32012
[16] Henkin, G.; Leiterer, J., Theory of functions on complex manifolds, (1984), Birkhäuser Basel
[17] Hoffman, K., Banach spaces of analytic functions, (1962), Prentice Hall New York · Zbl 0117.34001
[18] King, J., The currents defined by analytic varieties, Acta math., 127, 185-220, (1971) · Zbl 0224.32008
[19] Lawrence, M.G., Polynomial hulls of rectifiable curves, Amer. J. math., 117, 405-417, (1995) · Zbl 0827.32012
[20] M. G. Lawrence, Polynomial hulls of sets of finite length in strictly convex boundaries
[21] Mergelyan, S.N., Uniform approximation to functions of a complex variable, Uspekhi mat. nauk, 7, 31-122, (1952) · Zbl 0059.05902
[22] Sarkis, F., CR-meromorphic extension and the non-embedding of the andreotti – rossi CR structure in the projective space, Preprint of Paris 6, 116, (1997)
[23] Stolzenberg, G., Uniform approximation on smooth curves, Acta math., 115, 185-198, (1966) · Zbl 0143.30005
[24] Val’skii, R.E., On measures orthogonal to analytic functions in \(C\)^{n}, Dokl. akad. nauk SSSR, 173, (1967) · Zbl 0156.07803
[25] Vitushkin, A.G., Analytic capacity of sets and problems in approximation theory, Dokl. akad. nauk, 22, 141-199, (1967) · Zbl 0157.39402
[26] Wermer, J., The hull of a curve in \(C\)^{n}, Ann. of math., 68, 550-561, (1958) · Zbl 0084.33402
[27] Wermer, J., Function rings and Riemann surfaces, Ann. of math., 67, 45-71, (1958) · Zbl 0081.32902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.