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

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.

Extensions of the Laurent Decomposition and the spaces $A^p(Ω)$

“…) then we can easily check that g = Φ( f ), so Φ is surjective. By the Open Mapping Theorem, Φ : C p+1 (T) → C p (T) is an isomorphism, and its inverse is given by the assignment g → f , f being as in (4).…”

“…we can prove that the two semi-norms for l = 1 are equivalent (they are equal for l = 0). This can be done for all l ≤ p, l ∈ N, by induction, see for instance [4] .…”

“…A function f in the intersection of the two spaces would have to be continuous on C and holomorphic on C \ ∂Ω. The boundary ∂Ω having zero continuous analytic capacity translates to the fact that such f is entire ( [4], [6]). Since lim z→∞ f (z) = 0 we have by Liouville's Theorem that f is identically zero.…”

Domains of Holomorphy

“…For finite dimensional holomorphy Montel's theorem towards generic results has also been used in the works of Paul M. Gauthier; see for instance [5]. Some generic results for particular choices of X(Ω) have already been obtained in [1], [2], [6].…”

“…1 and F is holomorphic and bounded on B 2 . Thus, f satisfies Definition 2.2.Let Ω ⊂ C d be open and connected and X = X(Ω) let be a set of holomorphicfunctions f : Ω → C; that is, X ⊂ H(Ω).Definition 3.1.…”