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In this document I develop a weight function theory of positive order basisfunction interpolants and smoothers. **In Chapter 1 the basis functions anddata spaces are defined directly using weight functions. The data spaces areused to formulate the variational problems which define the interpolants andsmoothers discussed in later chapters. The theory is illustrated using somestandard examples of radial basis functions and two classes of weight functionsI will call the tensor product extended B-splines and the central differenceweight functions. **In Chapter 2 I derive modified inverse-Fourier transformformulas for the basis functions and the data functions (native spaces) and touse these formulas to obtain bounds for the rates of increase of thesefunctions and their derivatives near infinity. **Chapter 3 shows how to provefunctions are basis functions without using the awkward space of test functions$S_{0,n}$ which are infinitely smooth functions of rapid decrease with severalzero-valued derivatives at the origin. Worked examples include several classesof well-known radial basis functions. **In Chapter 4 we prove the existence anduniqueness of a solution to the minimal seminorm interpolation problem. We thenderive orders for the pointwise convergence of the interpolant to its datafunction as the density of the data increases. **In Chapter 5 a well-knownnon-parametric variational smoothing problem will be studied with specialinterest in the order of pointwise convergence of the smoother to its datafunction. This smoothing problem is the minimal norm interpolation problemstabilized by a smoothing coefficient. **In Chapter 6 a non-parametric,scalable, variational smoothing problem will be studied, with special interestin its order of pointwise convergence to its data function. **In Chapter 7 westudy the bounded linear functionals on the data spaces.

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