Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. Suppose further that this map is smooth (not just pw smooth) on the polyhedron of a subcomplex $L \subset K$ (feel free to assume its also a submanifold). My question is, can I approximate my $f$ with a **smooth** map $g$ which is also arbitrarily close to $f$ in the **Lipschitz norm** and with $g|_{\tau (|L|)}=f|_{\tau (|L|)}$? Here I assume K is sitting in some Euclidean space whose distance I use to define the Lipschitz norm. Please feel free to add hypotheses as needed. **Edit**: For starters I probably need M to be compact.

I have been browsing Hirsch's "smoothings of PL manifolds" but I haven't found anything about this particular question. Nonetheless, I suspect the answer is yes and that the argument is probably a fairly standard convolution argument so maybe this is really a reference request for the most natural general formulation of this question and where I can find the details of its proof.

I just added the geometric topology tag. If you feel this isn't a gt question please feel free to remove it.