On non-quasianalytic classes of infinitely differentiable functions

This article investigates the connection between two positive logarithmically convex sequences $\{\widehat{M}_n\}$ and $\{M_n\}$, which define respectively the Carleman classes of functions infinitely differentiable on the set $J$ and sequences $\{b_n\}$ specifying the values of the function itself and all its derivatives at some point $x_0\in J$. The results obtained are more general than those previously known, and explicit constructions of the required functions are proposed with estimates for the norms of the functions themselves and their $n$ derivatives in the Lebesgue spaces $L_r(J)$, not only for the classical case $r=\infty$ but also for any $r\geqslant 1$. Obviously, $M_n\leqslant\widehat{M}_n$ is always observed. Here the sequences $\{\widehat{M}_n\}$, for which equality holds, are indicated and specific examples are given.