APPROXIMATION USING SCATTERED SHIFTS OF A MULTIVARIATE FUNCTION

作者：Devore Ronald^*; Ron Amos

来源：Transactions of the American Mathematical Society, 2010, 362(12): 6205-6229.

• Image/signal processing
• computation
• nonlinear approximation
• optimal approximation
• radial basis functions
• scattered data
• thin-plate splines
• surface splines
• approximation order

The approximation of a general d-variate function f by the shifts phi( - epsilon), epsilon is an element of Xi subset of R(d), of a fixed function phi occurs in many applications such as data fitting, neural networks, and learning theory. Whim Xi = hZ(d) is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem using Fourier techniques. However, in most applications, the center set Xi is either given, or can be chosen with complete freedom. In both of these cases, the shift-invariant setting is too restrictive. This paper studies the approximation problem in the case that Xi is arbitrary. It establishes approximation theorems whose error bounds reflect the local density of the points in Xi. Two different settings are analyzed. The first occurs when the set Xi is prescribed in advance. In this case, the theorems of this paper show that, in analogy with the classical univariate spline approximation, an improved approximation occurs in regions where the density is high. The second setting corresponds to the problem of nonlinear approximation. In that setting the set Xi can he chosen using information about the target function f. We discuss how to 'best' make these choices and give estimates for the approximation error.

• 出版日期2010-12