Error estimates of quasi-interpolation and its derivatives

Quasi-interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications. A notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations. For radial basis functions with Gaussian kernel, there have been more studies on the interpolation and quasi-interpolation on infinite grids. This paper investigates the approximation by quasi-interpolation operators with Gaussian kernel on the compact interval. The approximation errors for two classes of function with compact support sets are estimated. Furthermore, the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated. Finally, the numerical experiments are presented to confirm the accuracy of the approximations.

[1]  G. Lorentz.  Approximation of Functions , 1966 .

[2]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[3]  Holger Wendland,et al.  Kernel techniques: From machine learning to meshless methods , 2006, Acta Numerica.

[4]  Vladimir Maz'ya,et al.  On Quasi-Interpolation with Non-uniformly Distributed Centers on Domains and Manifolds , 2001, J. Approx. Theory.

[5]  V. Maz'ya,et al.  On approximate approximations using Gaussian kernels , 1996 .

[6]  John P. Boyd,et al.  Error saturation in Gaussian radial basis functions on a finite interval , 2010, J. Comput. Appl. Math..

[7]  Jian Ping Liu,et al.  Generalized Strang–Fix condition for scattered data quasi-interpolation , 2005, Adv. Comput. Math..

[8]  Feilong Cao,et al.  Global errors for approximate approximations with Gaussian kernels on compact intervals , 2010, Appl. Math. Comput..

[9]  Marcello Sanguineti,et al.  Complexity of Gaussian-radial-basis networks approximating smooth functions , 2009, J. Complex..

[10]  Lei Wang,et al.  An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice , 2009, Appl. Math. Comput..

[11]  Frank Müller,et al.  Error estimates for approximate approximations with Gaussian kernels on compact intervals , 2007, J. Approx. Theory.

[12]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[13]  Lei Wang,et al.  Asymptotic coefficients for Gaussian radial basis function interpolants , 2010, Appl. Math. Comput..

[14]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[15]  N. Sivakumar,et al.  On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian , 2008, J. Approx. Theory.