Effective shape functions for the Genera lized Finite Element Method
should reflect the available information on the solution. This
information is fuzzy because the solution is, of course, unknown, and,
typically, the only available information is the solution's inclusion
in various funct ion spaces. It is desirable to select shape functions
that perform robustly over a family of relevant
situations. Quantitativ e concepts of robustness are introduced and
discussed.  We show in particular that, in one dimension, polynomials
are robu st when the only available information consist in inclusions
in the usual Sobolev spaces. If some additional informa tion is
available, if, {\it e.g.,} the approximated function is constrained by
certain boundary conditions, then polynomials ma y perform
poorly---relative to the optimal shape functions for approximating
functions satisfying the boundary cond itions---and some other family
of shape functions should be used. This talk is based on joint work
with I. Babu\v{s} ka and U. Banerjee.