[Paper Review] Nowhere differentiable functions with respect to the position

K. Kavvadias, Konstantinos G. Makridis|arXiv (Cornell University)|Jan 17, 2017

Advanced Mathematical Modeling in Engineering2 references3 citations

TL;DR

This paper establishes a generic dichotomy for functions in the rational approximation space $ R(\Omega) $ on bounded and unbounded domains $ \Omega \subset \mathbb{C} $: either all generic functions in $ R(\Omega) $ are nowhere differentiable with respect to position on $ \partial\Omega $, meaning the difference quotient blows up at every boundary point, or no such function exists. The key result is that the set of such functions is either empty or $ G_\delta $-dense in $ R(\Omega) $, proven via Baire category theory and conformal mapping techniques.

ABSTRACT

Let $Ω$ be a bounded domain in $\mathbb{C}$ such that $\partial Ω$ does not contain isolated points. Let $R(Ω)$ be the space of uniform limits on $\overlineΩ$ of rational functions with poles off $\overlineΩ$, endowed with the supremum norm. We prove that either generically all functions $f$ in $R(Ω)$ satisfy % $$ \limsup_{\substack{z o z_0 z \in \partial Ω}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$ for every $z_0 \in \partial Ω$ or no such function in $R(Ω)$ meets this requirement. In the first case, the generic function $f \in R(Ω)$ is nowhere differentiable on $\partial Ω$ with respect to the position. We give specific examples where each case of the previous dichotomy holds. We also extend the previous result to unbounded domains.

Motivation & Objective

• To investigate the generic behavior of functions in $ R(\Omega) $ with respect to differentiability with respect to position on the boundary $ \partial\Omega $.
• To determine under what conditions the difference quotient $ \left| \frac{f(z)-f(z_0)}{z-z_0} \right| $ diverges as $ z \to z_0 \in \partial\Omega $.
• To extend results on nowhere differentiable functions from real parameterization to complex position-based differentiability.
• To establish a dichotomy: either the set of such functions is $ G_\delta $-dense in $ R(\Omega) $, or it is empty.
• To provide examples where each case of the dichotomy holds, including both bounded and unbounded domains.

Proposed method

• Use of Baire category theorem to analyze genericity in the Banach space $ R(\Omega) $, equipped with the supremum norm on $ \overline{\Omega} $.
• Application of conformal mappings $ \phi $ to transfer the problem from a general domain $ \Omega $ to the unit disk $ D $, preserving analytic structure.
• Construction of sequences $ z_n \to z_0 $ on $ \partial\Omega $ such that the difference quotient diverges, using properties of holomorphic functions and their derivatives.
• Definition of the set $ S(\Omega, J) $ as the set of functions $ f \in R(\Omega) $ satisfying $ \limsup_{z \to z_0, z \in J} \left| \frac{f(z)-f(z_0)}{z-z_0} \right| = +\infty $ for all $ z_0 \in J $.
• Extension to unbounded domains via the space $ \widetilde{R}(\Omega) $, defined as uniform limits of rational functions on compact subsets of $ \overline{\Omega} $, with a Fréchet topology.
• Use of the Osgood–Carathéodory theorem to ensure boundary homeomorphisms for Jordan domains, enabling transfer of results from the disk to more general domains.

Experimental results

Research questions

• RQ1 Under what conditions is the set of functions in $ R(\Omega) $ with divergent difference quotient at every boundary point $ z_0 \in \partial\Omega $ non-empty?
• RQ2 Is the set of such functions generic in $ R(\Omega) $, and if so, in what topological sense?
• RQ3 Can the phenomenon of nowhere differentiability with respect to position be extended from the unit disk to arbitrary domains, including unbounded ones?
• RQ4 Are there domains where no function in $ R(\Omega) $ exhibits this divergence, despite the boundary being perfect?
• RQ5 How do conformal mappings and analytic continuation affect the differentiability behavior on the boundary?

Key findings

• For bounded domains $ \Omega $ with $ \partial\Omega $ containing no isolated points, the set $ S(\Omega, J) $ is either empty or $ G_\delta $-dense in $ R(\Omega) $, establishing a generic dichotomy.
• In the case where $ S(\Omega, J) $ is non-empty, the generic function $ f \in R(\Omega) $ is nowhere differentiable with respect to position on $ \partial\Omega $, as the difference quotient diverges at every boundary point.
• An example is provided where $ \Omega = D \setminus [0, \frac{1}{2}] $, and $ S(\Omega, \partial\Omega) = \emptyset $, due to holomorphic extendability across the slit, implying finite derivative limits.
• Another example with $ \Omega = \{ z : \operatorname{Re}(z) > 0 \} \setminus [1, \infty) $ also yields $ S(\Omega, J) = \emptyset $, showing the phenomenon can fail even for unbounded domains.
• For Jordan domains $ \Omega $ with $ \phi' \neq 0 $ on the boundary, $ S(\Omega, J) \neq \emptyset $, and the set is $ G_\delta $-dense, extending known results from the unit disk.
• The result extends to unbounded domains via the space $ \widetilde{R}(\Omega) $, where the same dichotomy holds: $ S(\Omega, J) $ is either empty or $ G_\delta $-dense in $ \widetilde{R}(\Omega) $.