Polynomial approximation, local polynomial convexity, and degenerate CR singularities
Copyright policy )Polynomial approximation, local polynomial convexity, and degenerate CR singularities
We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.
degenerate CR singularity, \(q\)-convexity, \(q\)-concavity, Polynomial approximation, 32E20, 32F05, Approximation in the complex plane, Approximation by polynomials, uniform approximation, 516, FOS: Mathematics, polynomial convexity, Complex Variables (math.CV), Polynomial convexity, rational convexity, meromorphic convexity in several complex variables, Mathematics - Complex Variables, Functional Analysis (math.FA), Mathematics - Functional Analysis, Banach algebras of continuous functions, function algebras, Polynomially convex, 30E10, 32E20, 46J10, polynomial approximation, failure of polynomial convexity, CR singularity, locally polynomially convex compact, Mathematics, Analysis
degenerate CR singularity, \(q\)-convexity, \(q\)-concavity, Polynomial approximation, 32E20, 32F05, Approximation in the complex plane, Approximation by polynomials, uniform approximation, 516, FOS: Mathematics, polynomial convexity, Complex Variables (math.CV), Polynomial convexity, rational convexity, meromorphic convexity in several complex variables, Mathematics - Complex Variables, Functional Analysis (math.FA), Mathematics - Functional Analysis, Banach algebras of continuous functions, function algebras, Polynomially convex, 30E10, 32E20, 46J10, polynomial approximation, failure of polynomial convexity, CR singularity, locally polynomially convex compact, Mathematics, Analysis
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