A weight function theory of zero order basis function interpolants and smoothers

Details A weight function theory of zero order basis function interpolants and smoothers A weight function theory of zero order basis function interpolants and smoothers

2007-08-06

arxiv.org/abs/0708.0780v3

Mathematics

Numerical Analysis

VERS 1: 81pg, 3fig. Portable latex from Scientific Word 5.00 Build 2606. VERS 2: replaced vers 1 with consolidation of all the

Scientific paper

I develop a weight func theory of zero order basis func interpolants and smoothers.**Ch 1: the basis funcs and data spaces are defined directly using weight funcs. The data spaces are used to formulate the variational probs which define the interpols and smoothers of later chapters. The theory is illustrated using standard examples of radial basis funcs and a class of wt funcs I call the tensor product extended B-splines. **Ch 2: the theory of Ch 1 is used to prove the ptwise convergence of the minimal norm basis func interpolant to its data func'n and to obtain orders of converge. Data funcs are characterized locally as Sobolev spaces. Results of numeric experiments using the exten B-splines. **Ch 3: another set of error estims for basis func interpol. Some better results. Uses tempered distrib Taylor expansion of exp(i(a,x)). Includes all prev results. **Ch 4: 1-dim only, scaled hat basis func, data funcs have b'ded deriv's, order converg is 1, details in append. **Ch 5: a class of tensor prod wt funcs is introduced which I call the central difference wt funcs. They are closely related to the exten B-splines. The theory is applied to these wt funcs to obtain interpol converge results. **Ch 6: a non-param variational smoothing problem is studied with special interest in the ptwise converge of the smoother to its data func. This smoother is the min norm interpol stabilized by a smoothing coeff. **Ch 7: a non-param, scalable, smoothing prob is studied with special interest in its converge to the data func. We discuss the SmoothOperator software package which implements this algorithm. **Ch 8: Characterizes b'nded linear functionals on data space.

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