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Approximation results of Artin-Tougeron-type for general filtrations and for $C^r$-equations

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to $C^r$-rings, for $2\le r\le\infty$. As an auxiliary step we establish the surjectivity of the completion map for rings of $C^\infty$ functions, for a very broad class of filtrations.