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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Approximation of singularity sets with analytic graphs over the ball in $\mathbf {C}^2$
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by Marshall A. Whittlesey PDF
Proc. Amer. Math. Soc. 125 (1997), 3259-3265 Request permission

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.
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Additional Information
  • Marshall A. Whittlesey
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
  • Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email: mwhittle@math.brown.edu
  • Received by editor(s): May 17, 1996
  • Additional Notes: This work is part of the author’s Ph.D. thesis and was supported in part by the R. B. Lindsay Graduate Fellowship. The author would also like to express his appreciation for the guidance of his thesis advisor John Wermer
  • Communicated by: Eric Bedford
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3259-3265
  • MSC (1991): Primary 32E30, 32F15
  • DOI: https://doi.org/10.1090/S0002-9939-97-04077-X
  • MathSciNet review: 1415342