  Interpolation and approximation of functionals with conditionally positive
definite kernels is considered on sets of centers that are not determining for
polynomials. It is shown that polynomial consistency is sufficient in order to
define kernel-based numerical approximation of the functional with usual
properties of optimal recovery. Application examples include generation of
sparse kernel-based numerical differentiation formulas for the Laplacian on a
grid and accurate approximation of a function on an ellipse.
