Interpolation by Radial Basis Functions on Sobolev Space

Interpolation by translates of suitable radial basis functions is an important approach towards solving the scattered data problem. However, for a large class of smooth basis functions (including multiquadrics @f(x)=(|x|^2+@l^2)^m^-^d^/^2, m>d/2, 2m-d@?2Z), the existing theories guarantee the interpolant to approximate well only for a very small class of very smooth approximands. The approximands f need to be extremely smooth. Hence, the purpose of this paper is to study the behavior of interpolation by smooth radial basis functions on larger spaces, especially on the homogeneous Sobolev spaces.

[1]  M. Buhmann.  New Developments in the Theory of Radial Basis Function Interpolation , 1993 .

[2]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..

[3]  C. Chui,et al.  Approximation Theory VI , 1990 .

[4]  J. Yoon.  Approximation in Lp(Rd) from a space spanned by the scattered shifts of a radial basis function , 2001 .

[5]  Robert Schaback,et al.  Approximation by radial basis functions with finitely many centers , 1996 .

[6]  W. Madych,et al.  Multivariate interpolation and condi-tionally positive definite functions , 1988 .

[7]  C. D. Boor,et al.  Fourier analysis of the approximation power of principal shift-invariant spaces , 1992 .

[8]  Will Light,et al.  Advances in Numerical Analysis, Vol. II: Wavelets, Subdivision Algorithms, and Radial Basis Functions. , 1993 .

[9]  Jungho Yoon.  Computational Aspects of Approximation to Scattered Data by Using ‘Shifted’ Thin-Plate Splines , 2001, Adv. Comput. Math..

[10]  Zongmin Wu,et al.  Local error estimates for radial basis function interpolation of scattered data , 1993 .

[11]  W. Madych,et al.  Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation , 1992 .

[12]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[13]  J. Ward,et al.  Regarding thep-norms of radial basis interpolation matrices , 1994 .

[14]  David Levin,et al.  The approximation power of moving least-squares , 1998, Math. Comput..

[15]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .