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Keywords:
Carleman class; Sobolev space
Carleman class; Sobolev space
Summary:
For nonquasianalytical Carleman classes conditions on the sequences $\{\widehat{M}_n\}$ and $\{M_n\}$ are investigated which guarantee the existence of a function in $C_J\{\widehat{M}_n\}$ such that
u^{(n)}(a) = b_n, \quad\vert b_n\vert\le K^{n+1}M_n, \quad n = 0,1,\dots, \quad a\in J.
Conditions of coincidence of the sequences $\{\widehat{M}_n\}$ and $\{M_n\}$ 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 of the domain in a Sobolev space of infinite order is shown.
References:
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