Pathological Phenomena in Denjoy–Carleman Classes

Abstract

Abstract. Let C M denote a Denjoy-Carleman class of C ∞ functions (for a given logarithmically-convex sequence M = (M n )). We construct: (1) a function in C M ((−1, 1)) which is nowhere in any smaller class; (2) a function on R which is formally C M at every point, but not in C M (R); (3) (under the assumption of quasianalyticity) a smooth function on R p (p ≥ 2) which is C M on every C M curve, but not in C M (R p ).

Quasianalytic ultradifferentiability cannot be tested in lower dimensions

“…While many non-quasianalytic classes can be tested along non-quasianalytic curves in the same class [13], the analogous statement is false for quasianalytic classes even if the function in question is smooth. This was shown by Jaffe [10] for quasianalytic Denjoy-Carleman classes of Roumieu type. In [15] we overcame this problem by testing along all Banach plots in the class (i.e.…”

“…Composing f with the squared Euclidean norm in R n gives a function with the properties in the lemma. For details see [10].…”

Nonlinear Conditions for Ultradifferentiability

“…In fact, quasianalytic ultradifferentiability cannot be tested on quasianalytic curves (or lower dimensional plots) even if the function in question is known to be smooth ( [10,20]). Hence, we think that it is interesting that, combining our proof with a description of certain quasianalytic classes E {M} as an intersection of suitable non-quasianalytic ones (due to [16]), we obtain that these quasianalytic classes have property (D) in all dimensions (see Theorem 2.7 and also Remarks 3.3 and 4.4).…”

Composite quasianalytic functions

“…If , are Denjoy–Carleman classes, then , for all , if and only if (see [Thi08, § 1.4]); in this case, we write . For any given Denjoy–Carleman class , there is a function in which is nowhere in any given smaller class [Jaf16, Theorem 1.1].…”