Title: Zero Sets of Functions from Non-Quasi-Analytic Classes
Abstract: Any closed subset E of the real numbers R is the zero set of some C°°-function/.One can also specify the order d(s) of the zero of /at each element s of the set S of isolated points of E. The present note improves this result by showing that each nonquasi-analytic class C{M"] contains such functions.R. B. Hughes stated the foregoing interesting result in [1].The purpose of this note is to present a short proof of Hughes' theorem based on properties of H. E. Bray's construction presented on pp.79-84 of [2].Suppose that {M"}^=1 is a sequence of positive numbers; set px = Mx1 and pk=(Mk_x¡Mk), k>\.A C°°-function <f> on R belongs to the class C{Mn) if there is a positive number A. such that ||¿|L^/1, and Hm\\^A\Mk, *>í.The theory of C'-functions permits us to suppose, without loss of generality, that pk^.pk+x, k^.\.Then the Denjoy-Carleman theorem [3, p. 376] tells us that the class C{Mn) is a quasi-analytic class if, and only if,