Abstract
In this paper we re-examine Wendland’s strategy for the construction of compactly supported positive definite radial basis functions. We acknowledge that this strategy can be modified to capture a much larger range of functions, including the so-called missing Wendland functions which have been the subject of a recent paper by Schaback (Adv Comput Math 34:67–81, 2011). Our approach is to focus on a general integral representation of such functions and we will show how a careful evaluation of this integral leads to new closed form expressions for both Wendland’s original functions and the missing ones. The resulting expressions are easy to code and so provide the potential user with a quick way of accessing a desired example for a given application.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1964)
Askey, R.: Radial characteristic functions. Technical report no 1262, Mathematics Research Center, University of Madison-Wisconsin (1973)
Schaback, R.: Kernel based meshless methods. In: Lecture Notes for Taught Course in Approximation Theory. Georg-August-Universität Göttingen (2007)
Schaback, R., Wu, Z.: Operators on radial functions. J. Comput. Appl. Math. 73, 257–270 (1996)
Schaback, R.: The missing Wendland functions. Adv. Comp. Math. 34, 67–81 (2011)
Szmytkowski, R.: On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46, 231–260 (2009)
Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
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Communicated by Joe Ward.
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Hubbert, S. Closed form representations for a class of compactly supported radial basis functions. Adv Comput Math 36, 115–136 (2012). https://doi.org/10.1007/s10444-011-9184-5
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DOI: https://doi.org/10.1007/s10444-011-9184-5
Keywords
- Positive definite functions
- Compactly supported radial basis functions
- Hypergeometric functions
- Associated Legendre functions