Approximation of singularity sets with analytic graphs over the ball in 𝐂²

Whittlesey, Marshall

doi:10.1090/S0002-9939-97-04077-X

Approximation of singularity sets with analytic graphs over the ball in $\mathbf {C}^2$
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by Marshall A. Whittlesey

Proc. Amer. Math. Soc. 125 (1997), 3259-3265

DOI: https://doi.org/10.1090/S0002-9939-97-04077-X

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Abstract:

Let $h$ be a smooth function on the ball in C$^{2}$ whose gradient has length less than or equal to 1. We show that if $h$ is uniformly near an analytic function on every complex affine one-dimensional slice then it must be near some function analytic on the whole ball. We use this to show the following: a singularity set over the ball which is near the graph of a function $h$ with $|\nabla h|\leq 1$ must be near the graph of some analytic function over the ball.

References

Herbert Alexander and John Wermer, On the approximation of singularity sets by analytic varieties, Pacific J. Math. 104 (1983), no. 2, 263–268. MR 684289
Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99–109. MR 779607, DOI 10.1007/BF01455798
Hartogs, F., Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde, Acta Math. 32 (1909), 57–79.
Toshio Nishino, Sur les ensembles pseudoconcaves, J. Math. Kyoto Univ. 1 (1961/62), 225–245 (French). MR 147670, DOI 10.1215/kjm/1250525058
Oka, K., Notes sur les familles des fonctions analytiques multiform etc, Hiroshima Math. J., Series A 4 (1934), 93–98.
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703
Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255–260. MR 818455, DOI 10.1090/S0002-9939-1986-0818455-X
John Wermer, Maximum modulus algebras and singularity sets, Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), no. 3-4, 327–331. MR 592560, DOI 10.1017/S0308210500012233