Power substitution in quasianalytic Carleman classes
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Consider an equation of the form f(x) = g(xk), where k > 1 is an integer and f(x) is a function in a given Carleman class of smooth functions. For each k, we construct a non-homogeneous Carleman-type class which contains all the smooth solutions g(x) to such equations. We prove that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions.
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This work was started during the visit of the authors to IMPAN, Warsaw, in the framework of workshop “Contemporary quasianalyticity problems”. The authors thank Misha Sodin for encouragement and useful remarks regarding the presentation of this paper.
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