Polynomial approximation, local polynomial convexity,and degenerate CR singularities

Bharali, Gautam (2006) Polynomial approximation, local polynomial convexity,and degenerate CR singularities. In: Journal Of Functional Analysis, 236 (1). 351 -368.

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Abstract

We begin with the following question: given a closed disc (D) over bar subset of C and a complex-valued function F is an element of C((D) over bar), is the uniform algebra on (D) over bar generated by z and F equal to C((D) over bar)? When F F is an element of C-1 (D), this question is complicated by the presence of points in the surface S := graph((D) over bar)(F) that have complex tangents. Such points are called CR singularities. Let p is an element of S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.

Item Type:	Journal Article
Publication:	Journal Of Functional Analysis
Publisher:	Elsavier
Additional Information:	Copyright of this article belongs to Elsavier.
Keywords:	CR singularity;Polynomial approximation;Polynomially convex.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	06 Apr 2009 09:53
Last Modified:	19 Sep 2010 04:58
URI:	http://eprints.iisc.ac.in/id/eprint/17775

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