Abstract
A remarkable theorem of Joris states that a function f is C^infty if two relatively prime powers of f are C^infty . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.
Highlights
Recently Thilliez [30] showed that Joris’s result carries over to Denjoy– Carleman classes of Roumieu type E{M}. These are ultradifferentiable classes of smooth functions defined by certain growth properties imposed upon the sequence of iterated derivatives in terms of a weight sequence M
By extracting the essence of Thilliez’s proof, we show in this paper that a broad variety of ultradifferentiable classes has a division property equivalent to Joris’s result
E[M], ultradifferentiable classes defined by weight matrices of Roumieu and Beurling type (Theorem 4.2)
Summary
A remarkable Theorem of Joris [11, Théorèm 2] states: if f : R → R is a function and p, q are relatively prime positive integers, f p, f q ∈ C∞ ⇒ f ∈ C∞. Recently Thilliez [30] showed that Joris’s result carries over to Denjoy– Carleman classes of Roumieu type E{M}. These are ultradifferentiable classes of smooth functions defined by certain growth properties imposed upon the sequence of iterated derivatives in terms of a weight sequence M (which in view of the Cauchy estimates measures the deviation from analyticity). Suppose that p1, p2 are relatively prime positive integers and f p1 , f p2 ∈ S. Since two consecutive integers are relatively prime, the converse holds.
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