Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

# Spherical scattered data quasi-interpolation by Gaussian radial basis function

## Abstract

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasi-interpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.

This is a preview of subscription content, log in to check access.

## References

1. [1]

Boyd, J. P., Error saturation in Gaussian radial basis function functions on a finite interval, J. Comput. Applied Math., 234, 2010, 1435–1441.

2. [2]

Boyd, J. P. and Wang, L., An analytic approximation to the cardinal functions of Gaussian radial basis functions on a one-dimensional infinite uniform lattice, Appl. Math. Comput., 215, 2009, 2215–2223.

3. [3]

Buhmann, M. D., Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, Vol. 12, Cambridge University Press, Cambridge, UK, 2003.

4. [4]

Cao, F. L., Guo, X. F. and Lin, S. B., L p error estimates for scattered data interpolation on spheres, Numerical Functional Analysis and Optimization, 32(12), 2011, 1205–1218.

5. [5]

Cavoretto, R. and De Rossi, A., Fast and accurate interpolation of large scattered data sets on the sphere, J. Comput. Appl. Math., 234, 2010, 1505–1521.

6. [6]

Chen, D., Menegatto, V. A. and Sun, X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 131, 2003, 2733–2740.

7. [7]

Fasshauer, G. and Schumaker, L. L., Scattered data fitting on the sphere, Mathematical Methods for Curves and Surfaces II (M. Dælen, T. Lyche and L. L. Schumaker, eds), Vanderbilt University Press, Nashville, 1998.

8. [8]

Freeden, W., Gervens, T. and Schreiner, M., Constructive Approximation on the Sphere, Oxford University Press, New York, 1998.

9. [9]

Jetter, K., Stöckler, J. and Ward, J., Error estimates for scattered data interpolation on spheres, Math. Comput., 68, 1999, 733–747.

10. [10]

Kress, R., Numerical Analysis, Springer-Verlag, New York, 1998.

11. [11]

Kuang, J. C., Applied Inequalities, Shandong Science and Technology Press, Jinan, 2004 (in Chinese).

12. [12]

Le Gia, Q. T., Sloan, I. H. and Wendland, H., Multiscale analysis in Sobolev spaces on the sphere, SIAM J. Numerical Analysis, 48, 2010, 2065–2090.

13. [13]

Le Gia, Q. T., Sloan, I. H. and Wendland, H., Multiscale analysis for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere, Applied Computational Harmonic Analysis, 32, 2012, 401–412.

14. [14]

Le Gia, Q. T. and Tran, T., An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions, J. Complexity, 26, 2010, 552–573.

15. [15]

Levesley, J. and Sun, X., Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 133, 2005, 269–283.

16. [16]

Levesley, J. and Sun, X., Corrigendum to and two open questions arising from the article Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 138, 2006, 124–127.

17. [17]

Lorentz, G. G., Approximation of Functions, Holt, Rinehart and Winston, 1966.

18. [18]

Maźya, V. and Schmidt, G., Approximate Approximations, American Mathematical Society, Providence, 2007.

19. [19]

Müller, C., Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer-Verlag, Berlin, 1966.

20. [20]

Müller, F. and Varnhorn, W., Error estimates for approximate approximation with Gaussian kernels on compact intervals, J. Approx. Theory, 145, 2007, 171–181.

21. [21]

Narcowich, F. J., Sun, X. and Ward, J. D., Approximation power of RBFs and their associated SBFs: A connection, Adv. Comput. Math., 27, 2007, 107–124.

22. [22]

Narcowich, F. J., Sun, X. P., Ward, J. D. and Wendland, H., Direct and inverse sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., 7, 2007, 369–390.

23. [23]

Narcowich, F. J. and Ward, J. D., Scattered data interpolation on spheres: Error estimates and locally supported basis function, SIAM J. Math. Anal., 33, 2002, 1393–1410.

24. [24]

Schoenberg, I. J., Positive definite functions on spheres, Duke Math. J., 9, 1942, 96–108.

25. [25]

Wang, K. Y. and Li, L. Q., Harmonic Analysis and Approximation on the Unit Sphere, Science Press, Beijing, 2000.

26. [26]

Wendland, H., Scattered Data Approximation, Cambridge University Press, Cambridge, UK, 2005.

27. [27]

Wendland, H., Multiscale analysis in Sobolev spaces on bounded domain, Numerische Mathematik, 116, 2010, 493–517.

28. [28]

Wu, Z. M., Models, Methods and Theory of Scattered Data Fitting, Science Press, Beijing, 2007 (in Chinese).

29. [29]

Xu, Y. and Cheney, E. W., Strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 116, 1992, 977–981.

## Author information

Correspondence to Zhixiang Chen.

This work was supported by the National Natural Science Foundation of China (Nos. 61272023, 61179041).

## Rights and permissions

Reprints and Permissions

Chen, Z., Cao, F. Spherical scattered data quasi-interpolation by Gaussian radial basis function. Chin. Ann. Math. Ser. B 36, 401–412 (2015). https://doi.org/10.1007/s11401-015-0907-7

• Revised:

• Published:

• Issue Date:

### Keywords

• Scattered data
• Approximation
• Spherical Gaussian radial basis function
• Modulus of continuity

• 41A17
• 41A25
• 41A63