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For nonquasianalytical Carleman classes conditions on the sequences $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ are investigated which guarantee the existence of a function in $C_J\left\{{\widehat{M}}_n\right\}$ such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary...