Multivariate quasi-interpolation in L^p(ℝ^d) with radial basis functions for scattered data

Abstract

In this paper, quasi-interpolation for scattered data was studied. On the basis of generalized quasi-interpolation for scattered data proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], we have developed a new method to construct the kernel in the scheme by the linear combination of the scales, instead of the gridded shifts of the radial basis function. Compared with the kernel proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], the new kernel, which is still a radial function, possesses the feature of polynomial reproducing property. This opens a possibility for us to propose a different technique by obtaining a higher approximation order of the convergence.

Keywords:

• quasi-interpolation
• scattered data approximation
• polynomial reproducing
• radial basis function

2000 AMS Subject Classification :

• 41A63
• 41A25
• 65D10

Acknowledgements

The authors thank the referees for their valuable comments, both for mathematics and English formulation, that helped to complete the final version. This study was supported by NSFC No. 10125102, NBRPC 973-2006CB303102.