[Paper Review] Non extendability from any side of the domain of definition as a generic property of smooth or simply continuous functions on an analytic curve

E. Bolkas, Vassili Nestoridis+1 more|arXiv (Cornell University)|Nov 27, 2015

Analytic and geometric function theory6 references8 citations

TL;DR

This paper establishes that non-extendability of smooth or continuous functions from either side of an analytic curve is a generic property in various function spaces. Using Fourier analysis, Cauchy transforms, and complex analytic methods—including Montel’s theorem and Baire category arguments—it proves that, generically, functions in $ C^∞(T) $, $ C^k([0,1]) $, and $ A^\infty(\Omega) $ are nowhere analytically extendable across the boundary, even when parametrized by arc length.

ABSTRACT

In this article we show that extendability from one side of a simple analytic curve is a rare phenomenon in the topological sense in various spaces of functions. Our result can be proven using Fourier methods combined with other facts or by complex analytic methods and a comparison of the two methods is possible.

Motivation & Objective

• To establish that non-extendability from either side of an analytic curve is a generic property in function spaces such as $ C^\infty(T) $, $ C^k([0,1]) $, and $ A^\infty(\Omega) $.
• To demonstrate that the set of nowhere analytic functions is a dense $ G_\delta $ subset in these spaces, strengthening prior results on $ C^\infty([0,1]) $.
• To compare and unify two approaches: Fourier analysis with Borel’s theorem and Michael’s selection theorem, versus complex analytic methods using Hardy spaces and Montel’s theorem.
• To show that non-extendability is preserved under conformal mappings and that arc length parametrization does not alter the analyticity properties of boundary functions.
• To prove that for domains $ \Omega $ bounded by finitely many analytic Jordan curves, functions in $ A^\infty(\Omega) $ are generically nowhere real analytic on the boundary.

Proposed method

• Utilizes the Cauchy transform to decompose $ C^\infty(T) $ into $ A^\infty(D) $ and a complementary space $ Y $, enabling analysis of boundary behavior.
• Applies Baire’s category theorem and the Baire residual set argument to show that the set of non-extendable functions is dense and $ G_\delta $ in $ C^\infty(T) $ and $ C^k([0,1]) $.
• Employs Borel’s theorem to construct $ C^\infty $ functions with arbitrary jet sequences at a point, enabling local control of non-extendability.
• Uses Michael’s selection theorem to continuously assign $ C^\infty $ functions to sequences of derivatives, ensuring continuity in the construction.
• Applies a localized criterion for non-extendability: a holomorphic function on $ D $ is not extendable near $ 1 \in \partial D $ iff $ R_\zeta \leq |\zeta - 1| $ for all $ \zeta \in D $ (or a dense subset clustering to 1).
• Uses Montel’s theorem in the complex analytic approach to extract uniformly convergent subsequences of holomorphic functions, ensuring continuity of the limit.

Experimental results

Research questions

• RQ1 Is non-extendability from one or both sides of an analytic curve a generic phenomenon in $ C^\infty $ and continuous function spaces?
• RQ2 Can the non-extendability of functions on the unit circle $ T $ be extended to subarcs, segments, or the real line via conformal mapping?
• RQ3 How do Fourier methods compare to complex analytic methods in proving generic non-extendability, and in what ways can they be unified?
• RQ4 Does the choice of parametrization (e.g., arc length vs. conformal) affect the genericity of non-extendability on analytic boundaries?
• RQ5 Is the set of functions in $ A^\infty(\Omega) $ that are nowhere real analytic on $ \partial\Omega $ a dense $ G_\delta $ subset when $ \Omega $ is bounded by finitely many analytic Jordan curves?

Key findings

• The set of nowhere analytic functions in $ C^\infty([0,1]) $ is itself a dense $ G_\delta $ subset, strengthening earlier results.
• In $ C^\infty(T) $, the set of functions non-extendable from either side of the unit circle is a dense $ G_\delta $ subset, meaning non-extendability is generic.
• For $ C^k([0,1]) $, $ k=0,1,2,\dots,\infty $, the set of functions non-extendable from either side of the interval is a dense $ G_\delta $ subset.
• The complex analytic method using $ H^\infty(D) $, Montel’s theorem, and radial limits yields the same results as the Fourier method but with simpler, more natural arguments.
• The localized non-extendability criterion—$ R_\zeta \leq |\zeta - 1| $ for all $ \zeta \in D $—is both necessary and sufficient for non-extendability near $ 1 \in \partial D $, enabling effective construction.
• For domains $ \Omega $ bounded by finitely many disjoint analytic Jordan curves, functions in $ A^\infty(\Omega) $ are generically nowhere real analytic on $ \partial\Omega $, regardless of parametrization (including arc length).