Approximation of piecewise linear function by smooth function

Question

I am reading a paper: Target enumeration via integration over planar sensor networks

And I would like to know if there is any reference which deals with approximating piecewise linear functions defined in $\mathbb{R}^n$ by smooth ones. Maybe that reference could cover the procedure, the errors... Even a reference for the case $n=2$ would be nice.

My question arises as I would like to apply some techniques from Morse theory but I have a piecewise linear interpolation of some counting functions.

Thanks in advance and any help would be appreciated

Answer 1

Note that piecewise linear functions defined on $\mathbb R^n$ for n>1 would have to be discontinuous. Consider n=2, the plane tiles wouldn't meet across grid cell edges. Unless you actually mean bi-linear patches.

Anyway, if you want to blend the local pieces smoothly into a globally smooth function, take a look at this paper that shows how to build a curvature continuous surface using tensor product B-splines and partition of unity.