Convergence of differentiable functions on closed sets and remarks on the proofs of the ``converse approximation lemmas'' (Q620005)

In KAM theory and other areas of analysis, one is often led to consider a decreasing sequence of open sets \(U_n\subset{\mathbb R}^d\) on which a function \(f_n\) is defined. Assuming that \(U_\infty=\bigcap_{n\geq 0}U_n\) is closed, one asks whether \(f=\sum_{n\geq 0}f_n\) is Whitney differentiable on \(U_\infty\). A direct verification of the Whitney conditions is often difficult in applications, hence the need for convenient sufficient conditions. In the present paper, the authors give examples showing that even if the series and its derivatives converge fairly quickly, \(f\) can fail to be Lipschitz. Then they present suitable conditions involving not only the speed of convergence of derivatives, but also the geometry of the sets \(U_n\).