Localization properties of basic classes of $C^{\infty }$-functions

Abstract:

Suppose that each of $\mathfrak {M} = C\{ {M_n}\}$ and $\mathcal {K} = C\{ {K_n}\}$ is a basic class of ${C^\infty }$-functions defined on $I = [0,2\pi ]$ and $\mathfrak {M}$ is not a subset of $\mathcal {K}$. Then it is shown that $\mathfrak {M}$ contains functions which are nowhere locally in $\mathcal {K}$. One of the corollaries asserts that there are quasi-analytic functions which are nowhere locally analytic. It is also shown that many non-quasi-analytic classes contain functions which are nowhere locally quasi-analytic.