MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 167-172 (1999)

Some classes of infinitely differentiable functions

G. S. Balashova

G. S. Balashova, Department of Mathematics, Power Engineering Institute, Krasnokazarmennaja 14, 111 250 Moscow, Russia, e-mail:

Abstract: 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.

Keywords: Carleman class, Sobolev space

Classification (MSC2000): 26E10, 46E35

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