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

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