  This paper is devoted to the question of constructing a higher order Faber
spline basis for the sampling discretization of functions with higher
regularity than Lipschitz. The basis constructed in this paper has similar
properties as the piecewise linear classical Faber-Schauder basis except for
the compactness of the support. Although the new basis functions are supported
on the real line they are very well localized (exponentially decaying) and the
main parts are concentrated on a segment. This construction gives a complete
answer to Problem 3.13 in Triebel's monograph (see References [47]) by
extending the classical Faber basis to higher orders. Roughly, the crucial idea
to obtain a higher order Faber spline basis is to apply Taylor's remainder
formula to the dual Chui-Wang wavelets. As a first step we explicitly determine
these dual wavelets which may be of independent interest. Using this new basis
we provide sampling characterizations for Besov and Triebel-Lizorkin spaces and
overcome the smoothness restriction coming from the classical piecewise linear
Faber-Schauder system. This basis is unconditional and coefficient functionals
are computed from discrete function values similar as for the Faber-Schauder
situation.
