 98487 Enrico Valdinoci
 A remark on sharp estimates for high order nonresonant normal
forms in Hamiltonian perturbation theory
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Jul 3, 98

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Abstract. Using the method of majorants, we give an estimate of the rest for
the nonresonant actionangle normal forms
and exhibit a simple example suggesting the ``optimality''
of this estimate. Given an integer $k$, calling $\g$
the size of the small denominators up to order $k$,
we prove that the $k^{\mbox{th}}$ order remainder
is approximatively bounded by
$O(\e_0^{k})$ with $\e_0=O(\g^2/k)$.
Thus, if we disregard the dependence of $\g$ upon $k$, we obtain
a rest bounded by $({\mbox{const}}\;k)^k$.
These estimates are conjectured to be optimal: to support this idea we
present a simplified model problem with no small denominators,
formally related to the above calculations: this example exhibits a
factorial divergence.
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