  One of the basic principles of Approximation Theory is that the quality of
approximations increase with the smoothness of the function to be approximated.
Functions that are smooth in certain subdomains will have good approximations
in those subdomains, and these {\em sub-approximations} can possibly be
calculated efficiently in parallel, as long as the subdomains do not overlap.
This paper proposes a class of algorithms that first calculate
sub-approximations on non-overlapping subdomains, then extend the subdomains as
much as possible and finally produce a global solution on the given domain by
letting the subdomains fill the whole domain. Consequently, there will be no
Gibbs phenomenon along the boundaries of the subdomains. Throughout, the
algorithm works for fixed scattered input data of the function itself, not on
spectral data, and it does not resample.
