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On the successive derivatives of functions in a quasianalytic class
G. V. Badalyan
Abstract:
In this paper we investigate the question of the signs in the sequence $\{(-1)^n\varphi_n(u)\}$, where $\varphi_0(u)=\varphi(u)$, $\varphi_1(u)=\varphi'(u)$, $\dots$,
$$
\varphi_{k+1}(u)=\varphi{k+1}(u)_\gamma=\biggl(\frac{\varphi_k(u)}{u^{\gamma_k-\gamma_{k-1}-1}}\biggr)', \quad k=1,2,\dots,
$$
$0=\gamma_0<\gamma_1\leqslant\gamma_2\leqslant\dots\leqslant\gamma_n\leqslant\dots\to\infty$, when the real function $\varphi(t)$ belongs to a certain quasianalytic class in the sense of Carleman (according to the classification suggested by the author). A particular corollary of the result given in the paper is the correctness of Borel's hypothesis that there cannot exist a quasianalytic function $f(x)$ all of whose derivatives are positive at a given point in the domain of quasianalyticity of the function, except when the function is analytic.
Received: 10.02.1967
Citation:
G. V. Badalyan, “On the successive derivatives of functions in a quasianalytic class”, Math. USSR-Izv., 2:4 (1968), 745–779
Linking options:
https://www.mathnet.ru/eng/im2492https://doi.org/10.1070/IM1968v002n04ABEH000663 https://www.mathnet.ru/eng/im/v32/i4/p788
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