  I develop a weight func theory of zero order basis func interpolants and
smoothers.**Ch1 Basis funcs and data spaces are defined using wt funcs. Data
(native)spaces are used to formulate the variational problems which define our
interpolants /smoothers. Introduce the tensor prod extended B-splines.**Ch2
Prove the p'twise convergence of the minimal norm basis func interpol to its
data func and obtain orders of converg. Data func spaces for the B-splines are
locally Sobolev spaces.**Ch3 Another set of error estims for basis func
interpol. Use distrib'n Taylor expansion of exp(i(a,x)).**Ch4 Derive local
interpol errors for data funcs with bounded first derivs.**Ch5 Introduce class
of tensor prod wt funcs which I call the central diff wt funcs - related to the
B-splines. Apply theory to these wt funcs to obtain interpol converge results.
The data func spaces are locally Sobolev spaces. **Ch6 A non-param variat
smoothing problem studied with special interest in converge of smoother to its
data func. This smoother is the min norm interpol stabilized by a smoothing
coeff.**Ch7: A non-parametric, scalable, smoothing problem shown to converge to
its data func. We discuss the software which implements these algorithms.**Ch8:
Characterizes bounded linear functionals on data space.**Ch9 Bilinear form used
to characterize the bounded linear functionals on the data spaces generated by
the wt funcs.**Ch10 We derive an upper bound for the deriv of the 1-dim
(scaled) hat basis func smoother assuming the data func has bounded derivs and
large supp wrt. the data region.**Ch11: In one dim only; the local data funcs
are assumed to have bounded derivs on the data region and consider a scaled hat
basis func. If the basis func has large enough supp wrt. the data region then
we show the order of converg of the interpol is 1. **Ch12: Exten ops derived
based on Wloka; assumes rectangle condit.
