A quasianalyticity property for monogenic solutions of small divisor problems

TL;DR

This paper investigates the quasianalytic properties of function spaces relevant to small divisor problems, demonstrating generalized analytic continuation for solutions under certain conditions.

Contribution

It introduces new quasianalytic function spaces for small divisor problems and shows conditions for generalized analytic continuation of solutions.

Findings

Quasianalytic properties hold for functions on specific compact sets.

Generalized analytic continuation is possible through the unit circle.

Conditions on the sets K_j enable continuation of solutions.

Abstract

We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K_j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K_j's.