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The classical lemma of Borel reads: any power series with real coefficientsis the Taylor series of a smooth function. Algebraically this means thesurjectivity of the completion map at a point, $C^\infty(\Bbb{R}^n)\twoheadrightarrow \Bbb{R}[[\underline{x}]]$. Similarly, Whitney extensiontheorem implies the surjectivity of the completion at closed subsets of$\Bbb{R}^n$.  For various applications one needs the surjectivity of completion for general$C^\infty$-rings and general filtrations. We establish the necessary andsufficient conditions for this surjectivity.  Moreover, we prove: any element of the completion admits a$C^\infty$-representative that is real-analytic outside of the locus ofcompletion, has any prescribed vanishing rate "at infinity", and the prescribedpositivity behaviour at the finite part. Alternatively, one can impose on thesmooth representative a set of (compatible) linear conditions.

Surjectivity of the completion map for rings of $C^\infty$-functions. (Whitney extension theorem for general filtrations)