A weight function theory of zero order basis function interpolants and smoothers
نویسنده
چکیده
In this document I develop a weight function theory of zero order basis function interpolants and smoothers. In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples of radial basis functions and a class of weight functions I will call the tensor product extended B-splines. In Chapter 2 the theory of Chapter 1 is used to prove the pointwise convergence of the minimal norm basis function interpolant to its data function and to obtain orders of convergence. The data functions are characterized locally as Sobolev-like spaces and the results of several numerical experiments using the extended B-splines are presented. In Chapter 3 a large class of tensor product weight functions will be introduced which I call the central difference weight functions. These weight functions are closely related to the extended B-splines and have similar properties. The theory of this document is then applied to these weight functions to obtain interpolation convergence results. To understand the theory of interpolation and smoothing it is not necessary to read this chapter. In Chapter 4 a non-parametric variational smoothing problem will be studied using the theory of this document with special interest in its order of pointwise convergence of the smoother to its data function. This smoothing problem is the minimal norm interpolation problem stabilized by a smoothing coefficient. In Chapter 5 a non-parametric, scalable, variational smoothing problem will be studied, again with special interest in its order of pointwise convergence to its data function. We discuss the SmoothOperator software (freeware) package which implements the Approximate smoother algorithm. It has a full user manual which includes several tutorials and data experiments.
منابع مشابه
A weight function theory of positive order basis function interpolants and smoothers
In this document I develop a weight function theory of positive order basis function interpolants and smoothers. In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples...
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