Abstract
In this paper, we describe a novel approach to classical approximation theory of periodic univariate and multivariate functions by trigonometric polynomials. While classical wisdom holds that such approximation is too sensitive to the lack of smoothness of the target functions at isolated points, our constructions show how to overcome this problem. We describe applications to approximation by periodic basis function networks, and indicate further research in the direction of Jacobi expansion and approximation on the Euclidean sphere. While the paper is mainly intended to be a survey of our recent research in these directions, several results are proved for the first time here.
Highlights
Two of the major developments during the last quarter of a century are the use of radial basis function (RBF) networks in learning theory, and wavelet analysis in computational harmonic analysis
A new feature here is the ability to approximate functions based on “scattered data”, that is, evaluations where one does not prescribe the location of the points where the function is to be evaluated. The analogues of these results in the context of periodic basis function networks are given in Sect
The following theorem shows that the coefficients of the expansions in (2.33) provide a complete characterization of local smoothness classes, analogous to the corresponding theorems in wavelet analysis
Summary
Two of the major developments during the last quarter of a century are the use of radial basis function (RBF) networks in learning theory, and wavelet analysis in computational harmonic analysis. A very classical example of this approach has xk ∈ [−1, 1], M is the class of all twice continuously differentiable functions, ◦ is the L2 norm on R, Lg = g , and we let δ → ∞ In this case, one recovers the cubic spline interpolant for the data. While interpolation by RBF networks as well as the use of extremal problems to obtain a functional relationship underlying a data are very popular methods, there are some disadvantages. A new feature here is the ability to approximate functions based on “scattered data”, that is, evaluations where one does not prescribe the location of the points where the function is to be evaluated The analogues of these results in the context of periodic basis function networks are given in Sect.
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