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On some quasi-analytic classes

Abdelhafed Elkhadiri

TL;DR

The paper investigates quasi-analytic function classes beyond the classical Denjoy–Carleman framework by emphasizing a monotonicity property: whenever a function's derivatives at a point are nonnegative, a neighborhood inherits nonnegativity of all derivatives. It develops an alternative, direct approach to Carleman-type non-surjectivity results for the Borel map, extending to new quasi-analytic classes constructed from sequences of integers using log-convex regularization and Newton polygon techniques. By formulating and proving a Generalized Taylor theorem for these integer-sequence classes, the authors establish rigidity properties analogous to the Denjoy–Carleman setting, including monotonicity-based non-surjectivity and unique continuation phenomena. The results broaden the scope of quasi-analyticity with practical criteria and provide tools applicable to o-minimal structures and related function classes in real analysis.

Abstract

Using the so called monotonicity property, we prove that the Borel mapping restricted to some quasi-anlytic classes is never onto.

On some quasi-analytic classes

TL;DR

The paper investigates quasi-analytic function classes beyond the classical Denjoy–Carleman framework by emphasizing a monotonicity property: whenever a function's derivatives at a point are nonnegative, a neighborhood inherits nonnegativity of all derivatives. It develops an alternative, direct approach to Carleman-type non-surjectivity results for the Borel map, extending to new quasi-analytic classes constructed from sequences of integers using log-convex regularization and Newton polygon techniques. By formulating and proving a Generalized Taylor theorem for these integer-sequence classes, the authors establish rigidity properties analogous to the Denjoy–Carleman setting, including monotonicity-based non-surjectivity and unique continuation phenomena. The results broaden the scope of quasi-analyticity with practical criteria and provide tools applicable to o-minimal structures and related function classes in real analysis.

Abstract

Using the so called monotonicity property, we prove that the Borel mapping restricted to some quasi-anlytic classes is never onto.

Paper Structure

This paper contains 8 sections, 17 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Quasi-analytic functions of a real variable.
  3. Quasi-analytic functions according to Denjoy's point of view.
  4. Proof of sufficiency of theorem 2.6
  5. Monotonicity property for quasi-analytic Denjoy-Carleman classes
  6. Quasi-analytic classes associated to a sequence of integers
  7. Monotonicity property for quasi-analytic classes associated to a sequence of integers
  8. Generalized Taylor's Theorem

Key Result

Theorem 2.1

Let $f$ be a $C^\infty$ function on the interval $[a,b]$. The function is completely determined in the whole interval $[a,b]$ by its value and the values of its derivatives in any point of $[a,b]$, if the series of positive terms: is divergent

Theorems & Definitions (32)

  • Theorem 2.1
  • Proposition 2.2
  • Proof 2.3
  • Proposition 2.4
  • Proof 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proof 2.9
  • Theorem 2.10: Mandelbrojt
  • ...and 22 more