# Article

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Keywords:
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:
[1] Mandelbroit S.: Adjoining Series. Regularization of Sequences. Applications. Izdat. Inostrannoj Literatury, Moskva, 1995. (In Russian.)
[2] Bang T: On quasi-analytiske funktioner. Thèse, København, 1946.
[3] Borel E.: Sur les fonctions d'une variable réelle indéfiniment dérivables. C.R. Acad. Sci. 174 (1922).
[4] Carleman T.: Les fonctions quasi-analytiques. Paris, 1926.
[5] Carleson L.: On universal moment problems. Math. Scand. 9 (1961), no. 2, 197-206. DOI 10.7146/math.scand.a-10635 | MR 0142012 | Zbl 0114.05903
[6] Wahde G.: Interpolation on non-quasi-analytic classes of infinitely differentiable functions. Math. Scand. 20 (1967), no. 1, 19-31. DOI 10.7146/math.scand.a-10815 | MR 0214976
[7] Mitiagin B.S.: On infinitely differentiable function with the values of its derivates given at a point. Dokl. Akad. Nauk SSSR 138 (1961), 289-292. MR 0130946
[8] Ehrenpreis L.: The punctual and local images of quasi-analytic and non-quasi-analytic classes. Institute for Advanced Study, Princeton, N. J., 1961. Mimeographed.
[9] Balashova G. S.: On extension of infinitely differentiable functions. Izv. Akad. Nauk SSSR, Ser. Mat. 51 (1987), no. 6, 1292-1308. (In Russian.) MR 0933965 | Zbl 0643.26015
[10] Balashova G. S.: Conditions for the extension of a trace and an embedding for Banach spaces of infinitely differentiable functions. Mat. Sb. 184 (1993), no. 1, 105-128. (In Russian.) MR 1211368
[11] Dubinskij Yu. A.: Traces of functions from Sobolev spaces of infinite order and inhomogeneous problems for nonlinear equations. Mat. Sb. 106 (148) (1978), no. 1, 66-84. (In Russian.) Zbl 0414.35028

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