許多類型的徑向基底函數都有一個額外的自由參數，隨著參數增長，這些徑向基底函數會將會趨近平坦，我們希望利用這些基底函數去製造出一個可以內插 N 個點的函數。在單一徑向基底函數的情況下，當我們在解決問題時，雖然展開式的係數 λ 會呈現發散的狀態，但目標函數最終會趨近一個 Lagrange 多項式。本文會利用前文 [1] 的方法，去探討多類型徑向基底函數的內插問題及其特性。

Many types of radial basis functions have an additional free parameter, and as the parameter grows, these radial base functions will be flatness. We want to use these radial basis functions to create a function which could interpolate N points. In the case of only one radial basis function, when we solve the problem, although the coefficients λ of the expansion will diverge, the objective function will eventually approach to a Lagrange polynomial. In this paper, we will use the analogous method in [1] to explore the interpolation problem of hybrid radial basis functions and its properties.

Contents
1 Introduction 1
2 RBF in 1-D 5
2.1 Some examples and the limit result in 1-D . . . . . . . . . . . 5
2.2 Conditions for convergence and limiting approximation . . . . 6
2.3 Proof of theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Hybrid RBF in 1-D 12
3.1 Conditions of convergence and limiting approximation for odd
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Proof of theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Conditions of convergence and limiting approximation for even
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Proof of theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Numerical results 30
List of Tables
1.1 Some examples of radial basis functions. . . . . . . . . . . . . 2
1.2 Coe cients of RBFs. . . . . . . . . . . . . . . . . . . . . . . . 3
4.1 Odd points case. . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Even points case. . . . . . . . . . . . . . . . . . . . . . . . . . 31

[1] R. E. Carlson and T. A. Foley. The parameter R2 in multiquadric inter-
polant. pages 29{42, 21(1991).
[2] Tobin A. Driscoll and Bengt Fornberg. Interpolation in the limit of in-
creasingly
at radial basis functions.
[3] C. A. Micchelli. Interpolation of scattered data: distance matrices and
conditionally positive de nite functions. pages 11{22, 2(1986).
[4] M. J. D. Powell. The theory of radial basis function approximation .
(1990).
[5] S. Rippa. An algorithm for selecting a good value for parameter c in radial
basis function interpolation. pages 11{22, 11(1999).
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