Some classes of infinitely differentiable functions. (English) Zbl 0936.26012

Balashova, G. S.

Math. Bohem. 124, No. 2-3, 167-172 (1999).

Summary: For nonquasianalytic 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 \[ u^{(n)}(a) = b_n, \quad |b_n|\leq K^{n+1} M_n, \quad n = 0,1,\dots , \quad a\in J. \] 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.

MSC:

26E10	\(C^\infty\)-functions, quasi-analytic functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Carleman class; Sobolev space